 As you can see the topics name is straight lines. So this is the first session of straight lines We will at least take three sessions to complete it. Okay. This is not a small chapter It has got a lot of aspects into it and all of them are very very basic Essential features of your coordinate geometry because without them. I think it will be very difficult for you to survive and Do well in coordinate geometry Okay, so this chapter you can say is the Platform is the basis or Is the foundation of your further learning of coordinate geometry coordinate geometry is a very big Vertical of mathematics. We are further going to talk about Five more chapters straight pair of straight lines circles Parabola ellipse hyperbola all of them would be under the coordinate geometry section So what are we going to learn in straight lines? First thing I'm going to start with is the prerequisites the prerequisites Which is actually your review of the Cartesian coordinate system a review of the Cartesian Coordinate system Cartesian coordinate system So mostly we'll be working with Cartesian coordinates only the one which was given by the French mathematician Reni This cart is so the the French mathematician was any Descartes, okay We are not going to talk much about polar coordinates, which was given to us by Newton But however, I'm sure all of you are aware of polar coordinates as well Okay, so under this review. We are going to first talk about the distance formula Okay, we all know distance formula. It's just a quick recap as you can see It's a review review means I'll be quickly revisiting distance formula Okay, so if you have two points on the Cartesian coordinate Okay, we all know how a Cartesian coordinate is made So we have two reference lines which are called the x-axis and the y-axis Suspectively which are at right angles to each other the point where the axis meet is called the origin So now when you refer to any point, let's say a point a which has got coordinate as x1 y1 What does x1 represent? X1 represents or you can say mod x1 represents. How far is your point from the y-axis? Or you can say x1 represents the directed distance Now views over directed distance Right a distance which has got a direction just like a vector quantity So x1 represents the directed distance of the point from the y-axis and y1 represents the directed distance of the point from the x-axis right, so if you have two points A and B Whose coordinates are as shown to you on the screen? What is the distance between them? What are the distance between them? How do you find it? very simple Let us say I construct a Right angle triangle over here. Let me say this is point M Okay, what would be the coordinates of M point remember M will share the same y coordinate as a and same x coordinate as B So this will be x2 y1 Okay, so if I ask you a simple question, what do you think is the distance of a m or what is the length m if I ask you So what are the length a m over here? You'll say, okay I know that a is at a distance of x1 directed distance x1 from the y-axis and M is at a directed distance x2 from the y-axis Correct, so if you know that there are two points whose distances from a fixed line is x1 and x2 Then what is the distance between the points themselves? You will say simple it is mod x2 minus x1 or mod x1 minus x2 Remember mod basically is important because I'm talking about the distance over here So mod x2 minus x1 Similarly, what would be the length Bm? Bm will say mod y2 minus y1 So Bm is mod y2 minus y1 Now you can use your Pythagoras theorem over here So Pythagoras theorem says AB square is equal to AM square plus Bm square Correct, so AM square means x2 minus x1 the whole square the moment you are squaring it mod Becomes irrelevant to us Y2 minus y1 square. So this is your AB Which is under root of this. This is the distance formula, which I am sure all of you know from class 9th onwards Okay, any questions here? Any questions easy? Okay, so if this is easy, I have a question for all of you. Let us solve this question Very simple question find the find the Find the coordinates of Find the coordinates of the circum center of the triangle Whose vertices are as given to you on the screen? Also find its radius also find its radius very simple question. Just based on this formula, which we have done So all of you please solve this and put forth your response on the chat box Yes, Rajaji Nagar people still haven't told me when is the exam starting? Okay, first of October. Oh, sorry. I was actually not looking at my chat. I'm so sorry. Okay First of October Gandhi jindis and second of October, right? So I'm sure everybody would have got the notification for the leaves that Centre Macadam is giving you finally So you don't have any classes from 18th to 25th of October Inclusive of both the days. So it the last class that will happen will be on 17th. If at all you have any class on 17th and the class after the Reopening will happen on 26 Okay, so 18 to 25th. No classes for you Of course, we'll give you some assignments. We'll give you some homeworks Okay What's there? Vacation also you're not allowing us to stay properly Okay Good Raghav Excellent. What about the center coordinates? Nobody has given me the center coordinate so far. So let us say This is your A. This is your B. This is your C This is not to the exact positioning of them in the coordinate axis. I'm just randomly drawing it Okay, but wherever required I would suggest you to draw your Diagrams as close to scale and as accurate as you can because many a times that also gives you a lot of hint Yeah, so this was a triangle. I hope you all know that the circum center is obtained by perpendicularly Bissecting the sides. Okay, this is your circum center given by S Very good to think sure kinshuk. I think so that's not correct. Just check your working kinshuk. Okay, very simple If you know that there's a point which is going to be the circum center It's distance from the vertices should all be the same, right? So s a Should be same as SB should be same as SC isn't it Yes, I know Okay, so s a will be equal to SB is equal to SC remember this will Result into two independent equations Please do not think that there are three equations coming from here because if you do SA equal to SB and you do SA equal to SC It will automatically imply SB equal to SC. Okay, so here you can get only two independent Equations from each other. So let's say I do SA equal to SB Now for that I need to assume this point. Let's assume it to be X comma Y But my advice is don't assume things to be X comma Y in coordinate geometry because When you're writing equations, you are also using X comma Y. So that will create a confusion Okay, since there is no, you know equations involved over here I mean, of course equations will come but there's no point or we don't have to state any kind of an equation in our answer We will we will we can take it to be X comma Y But my advice to you is don't take X comma Y as the name of any point. That's not advisable, right? Okay, so what is SA? SA would be under root of now Can I say SA is equal to SB will result into SA square is equal to SB square both are same thing So SA square will be X minus 5 whole square Y minus sorry Y plus 1 whole square SB will be SB will be X plus 1 whole square Y minus 5 whole square. Okay, so if you simplify this, you'll get minus 10 X Will get a plus 2 Y you will get 25 plus 1 here also will get 2 X You will get minus 10 Y and 25 plus 1 so 25 plus 1 will go off. I Remember, I'm not writing X square and Y square because I know they will get cancelled off So I always take a minimalistic approach. I write as less as possible because the more you write more mistakes you'll do Okay, so this will you guess this will give you 12 X is equal to 12 Y. That means X is equal to Y So this is my first condition that means the X and the Y coordinates, which is called abscissa and ordinate I hope you know X coordinate is called the abscissa Y coordinate is called the ordinate abscissa and ordinate would be equal Second thing I can use is let's say s s a is equal to sc That means s a square is equal to sc square So again, if I write the expression, I'll have X minus 5 whole square Y plus 1 the whole square is equal to SC square SC square is X minus 6 the whole square Y minus 6 the whole square Okay, so this will give you minus 10 X plus 2 Y Plus 26 this will give you minus 12 X minus 12 Y plus 36 plus 36, which is 72. Okay Let me see if I can drag up. Oh, I have reached the end of the screen. Okay. Anyways, I'll try to squeeze in over here So this will give you our 2 X plus 14 Y is equal to 46 Remember X and Y are equal. So 16 X is equal to 46. So X is equal to 23 by 8 Okay, so why will also be equal to 23 by 8? Because X and Y are equal. Okay, so this was your second equation. I'm solving 1 and 2 simultaneously So your center coordinates would be 23 by 8 comma 23 by 8 So what is going to be the radius radius basically is the distance of this point from any one of the coordinates Okay, so let's say I want to find out the radius radius would be under root of 23 by 8 minus 6 the whole square and The same thing repeated once again So that's actually a good thing for us because we can write it as 23 by 8 minus 6 times root 2 That's going to give you more of it. Sorry more of it. That's going to give you That's going to give you 25 by 8 if I'm not mistaken. Yeah, 25 by 8 root 2 units Easy question. Everybody would have got it later on when we learn more things will solve it by different methods We'll find the equations of the you know Sides will find out the equations of their perpendicular bisectors and then we'll see where they meet and get our answer But as of now since we have only learned distance formula officially in today's class. I'm going to use this Okay Can you go down once again? Yes, sure. Sure. Sure. Is this fine? I showed them. I'm sorry sit down Left. Thank you so much. Let's go on to the next question Next question next question. Let's take next question If the line segment joining the points a comma B and C Comma D subtend an angle of theta at the origin find cos of theta find cos of theta So basically the question is pretty simple if you have a Point a comma B If you have a point B C comma D Okay, and if you connect it to origin, I'm not solving it by the way I'm just making a diagram for myself because later on I have to Make it. What is this angle theta that they're asking? What is this angle theta that they're asking? I'll put the poll on I'll put the poll on. Oh my god three answers and all of them have chosen different different options. Okay No using off slope concepts. So those who have done all these straight lines in your school Our slope concept is not required just by the use of distance formula. We need to solve it Okay, good. We'll check. Let me take your attendance also Okay, let's Finish this off in the next 30 seconds 14 of you have responded so far next 30 seconds Will close the poll 5 4 3 2 1 Please vote. Please vote Okay, okay Good enough data points. So 30 of you have voted and maximum vote has gone to both option B and C both option B and C Okay, so what I'm going to use here. I'm going to first connect B and A I'm going to connect B and A Now we all know we all know The cosine law We all know the cosine law cosine law cos of theta is given by The side OA square OB square Minus AB square by a two times OA OB by two times OA OB Correct so OA square OA square is a square plus B square OB square is C square plus D square Minus AB square AB square will be a minus C the whole square B minus D the whole square correct By two OA OA is under root of a square plus B square under root of C square plus D square Okay When you open the brackets, you realize a square will get cancelled B square will get cancelled C square will get cancelled D square will get cancelled. So what will we left off? You'll be left with two AC You will be left with two BD and down there. We have a two Which will get cancelled off with that two on the numerator. So if you cancel it off, you get AC plus BD upon A square plus B square under root of C square plus D square. So which option is correct? Which option is correct? AC plus BD option B B is the right option Okay, so Janta was correct, but More than the correct Janta wrong Janta was also there. Okay Anyways, so let's not talk about the section formula. Do you want to copy something? Would you scroll down? Okay section formula I'm sure you would have heard of it, right Section formula you have done it in class 10th, right? So in section formula, we basically learn the coordinates of points which divide the join of two points either internally or externally I Think more you would have done internally then externally division, right? Okay, so we'll talk about two types of division one is called internal division and the other is called an external division Other is called the external division Okay, so what is an internal division? So internal division is basically a case where join of two points Let's say A and B are two points The join of these two points are divided by or is divided by a point C In such a way that C lies in the line segment connecting and B C lies within the line segment connecting and B So this is what we say C is internally dividing the join of A and B Okay, so normally when this ratio is M is to N You have all learned this in your class 10th I believe that if the coordinates of A is x1, y1 and B is x2, y2 Coordinate of C would be M x2 N x1 by M plus N comma My 2 and y1 by M plus N Correct normally we do not need the exact value of M and M But we need its ratio correct for example if it is 2 is to 4 or 3 is to 6 or 5 is to 10 or 10 is to 20 it is all considered as 1 is to 2 only right So we normally need a simplest ratio of M and N which we normally call as a K Okay, now I would like you to understand one small thing over here very important Very very important. So first of all the formula is changed like this. This is something which you all know It's not a new thing to you. Okay. Now. What you to understand this here is that when you talk about a internal division This ratio K is actually taken as positive So for internal division, we take K as positive Why we take as positive is because K is what K is the ratio of AC is to CB Right, that is what you're calling as M is to N Isn't it? But if you see it is not the distance AC. It is the directed distance AC So if I'm going from A to C, I will take this direction. Isn't it? This is going to be my direction. Correct And if I'm going from C to B, this is going to be my direction. So since both the directions are in the same You know facing towards the same way or both are in the same direction We take this ratio to be a positive quantity Now why I'm telling you this is because Soon we will learn about external division. There will take K as negative. So I'll come to it. I'll come to it So this is done because AC and CB are Directed Lens Directed lens in the same direction In the same direction Okay, very very important. Okay, so for internal division When you when you let's say when you're solving a problem, let's say I gave you a question and I said In what ratio a point X divides the join of AB Okay, so normally what you do you assume the ratio to be K and proceed Let's say K comes out to be positive So the the positiveness of this quantity will indicate that it was a case of internal division It was a case of internal division. Okay on the other hand when you talk about external division External division is basically where the join of a and b Let's say this is my a and b if extended if Extended let's say I extend this further now. It can be extended extending in both the directions, right? You can extend in the direction of A also you can extend the direction of B also if extended There is a point C lying here, which will divide it in the ratio of M is 2n Are you getting my point? Right, but here this m by n will actually be taken as a negative quantity This will be taken as a negative quantity why because Here your K is actually a C as you can see a C are going this way Correct a to C is to CB CB is coming downwards see their opposite direction just like vectors Okay, so AC is to CB would be considered to be negative in nature because AC and CB are oppositely directed because AC and CB are Oppositely directed Guys, let me bring a very simple thing into your notice. You are now dealing with Analytical geometry you're now dealing with coordinate geometry where direction is also important We are not dealing with geometry Like in your childhood you were dealing with geometry where only the magnitude mattered How how you know big is a side or what is it? You know Length or what is the area but now you're dealing with coordinate geometry? So science will come into picture which will signify directions to you Are you getting my point? So here this ratio AC is to CB would be like to directed lens Which are in opposite direction and therefore the ratio AC by CB will become a negative value Are you getting my point? So let's say you are solving a question and you get the K value as negative Then it will indicate that it's a case of external division It's a case of external division Mind you whether you're talking about internal or whether you're talking about external ABC must be collinear for a case of division to happen. This is very important Why it is important? I'll give you a question on that in some time. So let me complete this So, yes, so what would be the coordinates of C the C coordinates you can write it as K X to let's say I consider this ratio to be K K X 2 plus X 1 by K plus 1 comma K Y 2 plus Y 1 by K plus 1 Now normally this K itself is a negative quantity So I'm not putting an extra negative sign in front of it as what will happen You will take K as negative also and you will put a negative sign also gone long problem will become wrong Okay, so formula is not going to change is just that K is a negative quantity and K cannot be minus 1 and K cannot be minus 1 So no change in the formula is just that K was positive here Okay, and K is negative over here, but it cannot be minus 1. Are you getting my point here? Any questions any concerns over here? Any questions and concerns over here? Okay, now a couple of things here many people ask me sir if K becomes negative How would I know it is towards the side of B or it is C is towards the side of A? So, how would you know? How would you know whether this C is towards B side or C is towards A side? Can somebody tell me that? Okay, if I give you a K, let's say I say K is minus 2 by 3 Was C towards the side of B or was it towards the side of A? Absolutely, Aniruddha. It was towards the side of A. So this is something very important So, please make a note of this if If your mod of K is greater than 1 then it is farther away from A Okay, then C is farther away from it farther away from A If mod of K is between 0 to 1 Then C is farther away from B farther away from B Okay Very important Now I would like you to do a small question based on this. I just wanted to test your basic understanding Can I give you a question small question anything that you would like to copy here? Oh, Archit is there. How are you Archit? Feeling better? All well Don't stress yourself. Please your exams are coming here. Okay So if you want you can take a rest our classes are always recorded I know you would always love to attend a live session because then you get to ask questions participate in the class process Okay, but if your health is not permitting do take a rest. No worries Okay, so I have a question for you Let's say a is 2 comma 3 B is 5 comma 0 and Let's say C is 1 comma 1 Okay, so there are three points whose coordinate is given to you find the Find the ratio in which Find the ratio in which C divides The join of AB C divides the join of AB Okay, simple question. I'm sure you would have done much more difficult question than this in your class 10th itself But I would like to see your response Find the ratio in which C divides the join of AB Okay, very good Rowan that is the point actually That's actually the point Okay, so so far so far Nobody has given me the right answer actually Okay, trust me actually very good that all of you have responded Right Raghav. That's the answer Actually, it was okay. Let's discuss. Let's discuss. I'll not disclose that the fact right now So let's say this is your a 2 comma 3 Guys is it lagging when you're seeing it. Is there a lag? It's fine, right? Okay, so let us say C point which is one comma one is dividing it in the ratio of case to one Okay, normally we assume case to one because if case positive It will be a case of internal division if case negative then it's a case of external division as of now I just assume it to be k comma case to one. Okay, so as per the formula the answer would be 5k 5k plus 2 by k plus 1 comma comma 5 into 0 Sorry k into 0 1 into 3 by k plus 1 This should be the coordinate of the point C Okay, and it is already given that it is 1 comma 1 Right, but the reason I gave this question was many people they only check with one of the Coordinates for example, they will compare the abscissa Okay, and then they will jump to a conclusion. Let's say I compare this and I jump to a conclusion that it is Minus 1 by 4 that means it's the case of external division of Ratio 1 is to 4 it being closer to a then be okay, but hold on hold on When you compare the Ordinate that is 3 by k plus 1 is equal to 1 you get the shock of your life And you see k value coming out to be 2 that means the ratio is also 2 is to 1 internally When there is an inconsistency in your answer When there is an inconsistency in your answer it actually is a way That the question wants to tell you or the answer wants to tell you that there is no division happening at all Okay, in fact ABC will make a triangle Remember for division to happen whether it is internal or inter external ABC must be collinear Okay, so since ABC ABC Are not collinear Collinear meaning meaning in the same line collinear It implies that see does not divide the join of a and b does not divide the join of a and b So it was a bluff question. It was a trick question. So there's no to answer the answer it There's no division happening at all. Why I gave this question is because 99% as you can see most of you made this mistake You just compare one don't have to compare one you have to compare both of them If both answers are coming out to be same, then that is the answer So you can state that as your ratio, but if both of them are giving you different values, okay It is just a way that the answer tries to tell you that hey, there is no division happening at all Don't even try it Okay No, just thought of course plotting is another way or more robust way to do it Is this fine any questions here any any concerns? Okay, let's take a small question on this It's a match the column question Let me put the poll on Very good guys again when when column type question comes take a minimalistic approach Okay, take a minimalistic approach means go for The factor which is bringing out the difference between the options Okay, for example, I would never go for the first one. I will never solve the first question because If let's say I get an S. Let's say if I get an S then there are three such options which say SSS Okay, so I have to differently go for the next one. Okay, so it's better to start with That's particular in a column one question We should bring about the distinction in the option pretty quickly. This is called minimalistic approach None. Okay, let's wrap this up in another 20 seconds Not many people have responded very good cinchin Let's check let's check. Okay, Akshat Cinchin has responded after a long time cinchin. Well, what happened cinchin busy kind of okay Okay, five four three two one vote Okay, Akshat has given a different answer. Okay Shell low will close the poll. I think come on come on if somebody's voting please do so come on Okay, let's adhere to our timelines Dear students, nothing is more powerful than time. Yeah So most of you have gone with see see for cat Let's check So as I told you since there is PPP coming here RR coming over here I would either go and solve I will either go and solve the fourth one or I will go and solve the second one Okay, that'll make my life. It's easy. Okay. Let's let me solve for the second one The ratio in which negative 2 comma negative 9 divides the line segment joining joining 1 comma 3 and 2 comma 7 Okay, so let's say this point is called The point C which is negative 2 comma negative 9. Let's let the ratio be case to 1 so 2k 2k plus 1 by k plus 1 is equal to negative 2 and And and and and and 7k plus 3 by k plus 1 is negative 9 is negative 9 Right, so let's see what comes out from here. So 2k plus 1 is minus 2k minus 2 Correct. So 4k is minus 3. So k is negative 3 by 4 here Also, 7k plus 3 is minus 9k minus 9 16 k is minus 12. So k is minus 3 by 4. Okay, so This ratio piece to k is Now they're not mentioned They have not mentioned whether they are okay, so this can either be seven or It could either be one, but one is not there, right? Because if I take their absolute value, it should be actually be one, right? Sorry, if I take their actual value, it should actually be one, but the thing it is seven So the one which has got s So s is only in the second one Okay, so option number a becomes correct for me Am I right? Because for two only option number a says S Okay, okay fine. Let's say I have an element of doubt. I'll go for the next one. I'll go for let's say the fourth one In what ratio or sorry p and p comma q divide the join of minus 1 comma 2 and 4 comma minus 5 in the ratio of 2 is to 3. Okay, so 2 into 4 2 into 4 3 into minus 1 By 5 should be equal to your p. So p comes out to be 1 Okay, and let's see what's q value so 3 into 2 3 into 2 2 into minus 5 By again 5 that gives me q value as as as as How much is how much does it come out to be? 3 into 2 2 into minus 5 6 minus 4 by minus 5 minus 4 by 5 So p plus q will become 1 by 5 p plus q will become 1 by 5 which option says 1 by 5 P says 1 by 5. So the one where your fourth one is mapped to Fourth one is mapped to to P So second one is mapped to to s. No second one these also not mapping. Oh External division my bad my bad. Oh, so sorry. Okay, so it was a case of an external division So I have to take one of them as a negative sign Sorry, sorry, sorry, sorry. So this will become a one now Okay, this will also become a one now and this will become a negative sign. Okay, everything will change my bad my bad so Yeah So p value will become minus 3 minus 8, which is minus 11 and this will become Minus 16 But I think then none of the options will match Pq divides externally the joint of minus 1 and 2 and minus 4 and minus 5 in the ratio of 2 is to 3 Okay, so what I did was I took one of them as negative Yeah Something strange is happening. So minus 2 into 4 Okay, 3 into minus 1 by minus 2 plus 3 which is going to be 1. That's correct So that gives me minus 11 as the value of this and this gives me minus 16 Minus 16 Okay, and they're asking me p plus q. Oh, yeah, yeah, yeah So this is minus 2 correct Yeah, so this will give you this will give you 10 Yeah, this will give you 16 correct now the sum is going to be 5 which of them says 5 Which of them says 5 are says 5 correct. So fourth one whichever has an R That will be the answer. So obviously is option number One again has a again has an R. So a is correct right and Only one person got this correct. Oh My god, most of you went for C Most of you went for C only one person got this correct Okay, anyways Is this understood? So basically what I did was I had to take minus 2 is 2 3 so minus 2 will come over it Okay So minus 2 and plus 3 is coming over it and minus 2 plus 3 will become 1 in the denominator giving you p value as minus 11 Q value as 5 so p plus sorry q value as 16 So p plus q will become a 5 which is matching with R. So can you check what you have done in second one? Okay, see second one what I've done in the second one. I've just assumed the ratio to be anything So I get minus 3 to 4 Okay, so they basically meant because they could be two answers from it. It could be either be one Correct. Oh So now see You have taken it to be 3 is to minus 4 okay in that case your answer will be minus 4 plus 3 which is minus 1 Okay, then yes Then yes, option C will be correct. Okay. Janta was correct. Okay, correct Raghav. Correct. Correct. So I'll just Cross this and I'll take my list Okay, thank you So I was wondering how can so many people make a mistake Yeah, sure Shraddha. I'll scroll it to the right So here my p plus q value was actually coming out to be minus 1 which is your which is your Q option. So second is mapping with Q second is mapping with Q and Fourth one was mapping with R. So whichever said second one was Q and fourth one was R That was only C option was said that done copied Shraddha No, no, no, they actually even I thought so even I thought so but the question was actually asking you to Algebraically add P and Q along with their sign. Okay, so it could either be minus 3 plus 4 or it could be minus 4 plus 3 So minus 3 plus 4 will give you a 1 which is not there in your column number 2 So only possibility as Raghav rightly pointed out and I thank him for that It will be minus 4 plus 3 which is going to give you a minus 1 how? See 3 by minus 4 or minus 3 by 4. Does it make a difference? Does it make a difference Shraddha? Both are fine. No both are same things Correct. So if I say the ratio is minus 1 is to 2 or 1 is to minus 2 both are same thing Anyone of the M&N can be negative right because ratio has to be negative Okay. Yeah, it was slightly confusing question, but I think Most of you were correct Okay, guys now I'm going to a very serious type of a question. Okay We have all learned our critical points in a triangle in our properties of triangles chapter. Okay, so let me Name this concept as important points in a triangle In a triangle Okay, so we are going to first talk about centroid What's a centroid? Centroid is the meeting point of the medians Okay, so let's say I have Medians, what is median median is basically a line which connects a Vertex to the midpoint of the opposite side, isn't it? So let's say a BC then it'll connect to the midpoint of the opposite side. This is a median. Okay, so similarly M You can call it as P. They are all the midpoints of their respective sites The meeting point of the median is called the centroid Okay, we know that centroid divides the median in the ratio 2 is to 1 2 towards the vertex side 2 is to 1 2 towards the vertex side Similarly, this median G sorry the centroid G will divide CP in the ratio 2 is to 1 2 towards the vertex side This is also 2 is to 1. Okay, if you know the ratio Can we get the coordinates of G? Now, I'm sure you would have done this in your junior classes. So without much waste of time I will only do it. So let us say this point is x1 y1 a coordinate is x1 y1 b coordinate is x2 y2 C coordinate is x3 y3 Okay, the midpoint is going to be x2 plus x3 by 2 comma Why 2 plus y3 by 2? correct now since you know M coordinate Right, and you know a coordinate and you know this ratio is 2 is to 1 So what will be the g coordinate g coordinate? You will say it is simply 2 into 2 into x 2 plus x3 by 2 plus 1 into x1 1 into x1 by 2 plus 1 comma comma Same thing for the white two times y2 plus y3 by 2 plus 1 into y1 by 2 plus 1 Okay, if you simplify this this will give you x1 plus x2 plus x3 by 3 comma y1 plus y2 plus y3 by 3 I'm sure you would have done this derivation in class 10. Okay, so everybody knows centroid is the is the Point through which the center of mass of the triangular lamina will pass. Okay, if you generalize this you take any polygon and if you just do this Summation of all the x coordinate by n summation of all y coordinate by n that will always give you the center of mass of that You know polygon lamina Okay, so this is a point through which the center of mass of the triangular triangular lamina will pass this is known to everybody. Okay, now my question is can you Can you give me the coordinates of or can you prove that the coordinates of the in-center of a triangle? Let me make a triangle first of all. Let me extend this further Okay, how is an in-center formed in center is formed by internally bisecting these angles So let's say this angle is bisected This angle is bisected and This angle is bisected. Okay, they will meet at the in-center, which we call as I Okay, I would now like to I would not like you to work on How do you get the coordinates of the in-center given that the vertices coordinates are known to you? Remember we had Discuss this result in our properties of triangle, but we never got a chance to prove it and I also told you that I will prove it in the coordinate geometry chapter Okay, so they has come that we now prove this formula So prove that the in-center coordinate is given by this expression where is the length of BC small b is the length of AC and small c is the length of AB Would you like to try it out anybody? Okay. I'll also help you with it first Get the coordinates or get the ratio in which D divides the join of BC, which I think you already know Angle bisector theorem Correct. I hope you all know this theorem angle bisector theorem angle bisector theorem Who will tell me what is angle bisector theorem? Class 10th concept and I'm sure in autonomous you would have done it in a very very detailed fashion Yes, how would you like to try what's an angle bisector theorem anybody? Very good. Very good ananya. So angle bisector theorem says that if there is an internal angle bisector in a triangle Then it will cut at the opposite side in such a way that Bd is to DC will be a b is to AC Absolutely Shadda, very very good. This is something which is going to guide me to solve this question Correct. So can I say this ratio Bd is to DC here would be C is to C is to be correct. So this ratio is C is to be So if I know What is the coordinates of B and what are the coordinates of C and I also know this ratio Can I not get the coordinates of D? Okay, so please tell me the coordinates of D You'll say very simple sir. It will be C into X3 B into X2. So let me write it down B into X2 C into X3 by B plus C comma B into Y2 C into Y3 by B plus C Any doubt here any concerns here, please do let me know Please do let me know Okay, no problem with this Coordinates of D absolutely no problem Okay, now all of you please listen to this very carefully If you look at the triangle if you look at the triangle ABD If you look at the triangle ABD Correct. Can I say I divides or can I say like this? Can I say in triangle ABD AI is to ID Now look at the figure AI is to ID will be equal to BA is to BD By the same angle bisector theorem by the same What happened to my writing today? By the same angle bisector theorem correct I'm sure you would agree with this. There is no You know discrepancy in this logic Correct because if you see ABD Basically BI is like your internal angle bisector, isn't it? So if you think this to be a triangle then AI is to ID would be same as BA is to BD Am I right? Am I right? Okay. Now Can somebody tell me what is BD length because AB length or BA length is C What is BD length? Who will tell you what is BD length? AB is already B Sorry, AB is already C my bad. What is BD length? What is BD length? Somebody please type on the screen on that. Sorry, not on the screen. Please type in the chat box Don't write anything on the screen. Please What is BD length? Total length is a and This ratio is C's to B. So what is BD length? That's my question The full length is a and This BD is to DC C's to B. Let me draw it So let's say there is there's a line whose length is a and There is a point which divides it in the ratio C is to be Okay, my question is what is BD length here? What is BD length? Ah, correct So it is going to be C by Yeah, BD length will be equal to C by B plus C times a am I right? C by B plus C times a correct No problem Raghav Understood. No, but this is let's say this is one is to two and this whole length is five Let us say in and I ask you what is BD length? What will you say? You'll say one third of five Isn't it that's what you're going to say. No, so this is C by B plus C into the length a Correct. So this makes it B plus C by a am I right? Am I right? Everybody's happy till here because I don't want to proceed further if you're not happy till here clear Ritu good Everybody first write clear on your chat box, then only I'm going to proceed Don't keep any doubt Why why a Prakul is asking why because total length of BC was a no in a triangle The side Opposite to vertex a has a length small a by convention. We follow this correct Prakul Okay, now my life is pretty easy after I get this is because I know this ratio is Now B plus C is it to a This ratio is B plus C is it to a Okay, I'm writing it over a Do you know a coordinate? Yes, X1 Y1. Do you know D coordinate? Yes. We just now figured out Do you know the ratio in which I divides a and D? Yes B plus C is to a then what are we waiting for? Let's find out I coordinate simple as that. So from here. I can say Coordinate of I would be let me write it in some different color. So that I distinguish it from the so I coordinate would be B plus C times the X coordinate of D. So B plus C times X coordinate of D, which is this guy B X to C X 3 by B plus C Okay Plus a Times X1 by B plus C plus a similarly, I will do it for The Y coordinate B plus C B X to C X 3 by B plus C Plus oh Why am I writing X? Sorry, why? Okay, a Y1 by B plus C plus a Okay, if you simplify this you automatically would realize B plus C here and B plus C here will get cancelled off Same with this guy also and you will be left with A X1 now I'm writing it as a X1 B X2 C X 3 by a plus B plus C comma a Y1 B Y 2 C Y 3 by a plus B plus C Is that okay understood any question? If possible keep this in your mind Okay, now I would like you to do this as a homework question. I will not do it right now for you Let us say there is a triangle. I'll take a Very thin triangle. I don't want to make a thick one Okay, we all know that if you bisect this angle, let me use my tools Let's say if you bisect this angle Okay, let's say this angle is same as this angle. Okay, so that angle is internally bisected And let us say Let me extend this further And let us say I externally bisect this angle. Okay, so what I've done this angle is This angle I've externally bisected it then they will meet at a point I one Which is basically called as one of the X centers Okay, where I one coordinate is given by minus a X 1 B X 2 C X 3 by minus a plus B plus C comma comma Minus a Y 1 B Y 2 plus C Y 3 by minus a plus B plus C Okay, please prove this prove this for your homework Okay Now, I'm sure you would have got a trend here if I had drawn the same thing opposite to B That is called I to you all know What would be I to coordinate? Okay, I'll write it down over here. No need to prove all of them. This one is sufficient I to coordinate would be a X 1 minus B X 2 C X 3 by a minus B plus C comma a Y 1 minus B Y 2 C Y 3 by a minus B plus C Okay, and I 3 would be a X 1 B X 2 minus C X 3 by a plus B minus C and a Y 1 plus B Y 2 minus C Y 3 by a plus B minus Okay, let's prove one of them. I think I 1 is sufficient enough. This is just for your knowledge So there is a trend here in the formula if it is an in-center You'll all have plus plus plus on the top If you have an X center opposite to a then only a will have a negative sign Of course both in the numerator denominator and also in the X and Y coordinate both Okay, if you're drawing X center opposite to B, there'll be a minus sign in B And if you're drawing an X center opposite to C, there'll be a minus sign in C So just prove one of them You're not seeing coordinate geometry. Let coordinate geometry start can shoot you will have to maintain a formula list Okay, can you move on to the next question? Actually, I would like to take that as a question now then as a theory Done copied done Okay Next thing that you're going to talk about is your circum center is your circum center Now I'm going to give this as a question to you. Let's let's take this as a question Let us say you have you have a triangle. Okay. Let's say this is a circum center. This is your x3 y3 So a b c is a triangle whose coordinates of the vertices are as given to you s is the circum center prove that prove that the circum center coordinates are given by X1 sine 2a X2 sine 2b X3 sine 2c by Sine 2a plus sine 2b plus sine 2c and similarly for your white ordinate or ordinate of the circum center Y1 sine 2a y2 sine 2b Y3 sine 2c by sine 2a sine 2b plus sine 2c See this formula is completely impractical because you will hardly see any kind of an uses for this formula But let us take this as a question so that our understanding of section formula or for that matter our understanding of geometry is clear Okay, so let's take this as a question. How will you solve this? What is 2a 2b 2c a is basically this angle? This is angle is this angle is a this angle is b this angle is c. How do you do this question? Anybody has any idea? Looks scary right the formula looks very scary So how on earth I should get to a Sine 2a sine 2b and sine 2c. That's the first Thing that we need to sort out No, no, don't worry about the formula per se think it as if it's a question Okay, I'm not interested in knowing this formula neither. Do I endorse that you should know this formula? I am Interesting in solving it as a question. Let's say somebody has given me a question prove that this is going to be a circum center coordinate Okay, I'll help you out. I know most of you would be clueless as of now What to do from where to start no idea? Okay, let me do one thing. Let me connect A to S and extend it further Okay, let's say it meets somewhere over here Okay, and let's say this point here is your point e Let's say okay at the same time. Let me drop Okay, first of all What is the coordinate of e? How will I find it out? This is my first step towards solving this question now in order to get the coordinate of e I should know in what ratio in what ratio e divides the join of Bnc right, so I need to know be by ec Okay, so what I'm going to do is just watch here I'm going to drop a perpendicular from s on to bd On to bc and let me call this as a d. Okay no, no, no, no He is not the midpoint of bc He's not the midpoint of bc I mean he may not be the midpoint of bc. I'm not saying will not may not a Now listen to this, I would make it very very clear, let me make a triangle over here and let me make a triangle over here also, can I say all of you please listen to this, can I say B is to EC is nothing but area of triangle SBE by area of triangle SEC, would you agree? Would you agree that area of triangle SEB by area of triangle SEC will be B is to EC, now why is that so is very easy, the reason for saying so is that BSEB and SEC will have the same perpendicular which is your SD length, so the ratio of the area would be same as the ratio of their basis which is actually BE is to EC, yes or no, correct, no similarity no similarity, see look at this triangle, look at this triangle which I am showing with a white boundary, this triangle and the one I am showing with a yellow boundary, of course SE would be overlapping in both the cases, these two triangles share the same perpendicular, correct which is SD, SD is the same perpendicular, correct, so can I say BE is to EC will be same as the ratio of the area of these two triangles, okay, now listen to this very carefully, can I say SA, SB and SC all will be equal to your R that is your circum, circle, radius, correct, yes or no, okay, now I would like you to tell me one small thing, very very small thing, what do you think is this angle, what do you think is the angle which I have shown to you on the figure, what is angle ESB, anybody, ESB, Gayatri tell me by what logic Gayatri it is 90 degree, by what logic it is 90 degree, radius to center always forms 90 degree, I don't know I have not learned such a geometry, why would radius to center will make a 90 degree, okay, anyways, let me make your life simpler, let me make your life simpler, as geometry is the father of all these coordinate geometry stuff, okay, let me draw the same triangle but now within a circle so that you are aware what you are doing, let's say this is A, B, C what is my question asking you, my question is asking you, let's say this is the center, okay, and I connect the center to A and extend it forward and I make this, what is this angle I am asking you, what is this angle I am asking you, now listen to this, if this is C, what is this angle, 2C, correct, so what is this angle, pi minus 2C my dear, none of you answered that, come on, right, it's 2C, at least you agree that this is 2C, angle subtended at the center is double the angle subtended by the same arc on the opposite segment, correct, yes or no, so this is 2C, so this remaining is pi minus 2C, okay, now what is the area of a triangle, area of a triangle is nothing but half into product of the two adjacent sides that is R and SE into sign of the angle sandwiched between them, so sign of the angle is sign of pi minus 2C, correct, by half SEC, now again let me ask you this question, what do you think is this angle, Y to 1, let me write R SE sign of what should I write over it, very good, can show pi minus 2B, very good, so it's very obvious that this guy will be 2B, just like this guy was 2C, so this remaining one is pi minus 2B, correct, very good can show, nice one, so R R gone, half half gone, SE SE gone, so it is sign of 2C by sign of 2B, correct, so what do I have, oh my god I figured out what ratio E divides BC, that is sign 2B is to sign 2C, isn't it, so this ratio is sign 2C is to sign 2B, correct, so I can get now my E coordinate, now this question mark is solved, B is to EC, B is to EC is sign 2C is to sign 2B, so now I'm equipped enough to get my E coordinate, what is my E coordinate, somebody please tell me fast, so my E coordinate would be sign 2B into X2, so I can write it like this, X2 sign 2B, sign 2C into X3, that is X3 sign 2C by sign 2B, oh now I realize why it's 2A, 2B's angles are occurring, oh, so this is, we are inching towards our solution, okay, let me first finish off writing this, are you all happy till the stage, then only I will proceed further, else I will not proceed, you have to write happy, you have to write happy in your chat box, not very happy, why can't you answer, no, you should be very happy, complicated, okay, so it's a complicated happiness, okay, not everybody is happy, yes it is difficult, come on, we are preparing for India's toughest exam, okay, if it is easy everybody will get it, okay, now this is just half the proof done, now we'll focus on, now we'll focus on what is AS is to SE, because then only I will get the ratio in which S divides the joint of A and E, so what I'll do, I'll make a fresh triangle, okay, everything is getting hodgepodge over here, so one fresh triangle, let me draw, okay, so this is my new triangle, I will make some quick lines here, okay, so this is your S, this was your E which you figured out, okay, don't worry about E right now, E is already safe with you, now the problem is I need AS is to SE, what is AS is to SE, if I find this my job is done, we'll wrap this problem if I get my E, S, how will I find S, so AS you will say sir AS so I know, AS is R, no doubt about it right, it is the circle radius only, how will I get SE, how will I get SE, let me again draw this D back, how will I get SE length, okay, so I'll help you out with this, now who will give me what is this angle, can't you guys talking about the angle, angle SD you're talking about, length D is 2P into 2C, how, and B is the angle my dear, how can length be in terms of angles, if I'm not right B is an angle right, yes sir if I'm not wrong B is an angle, how can length be in terms of angles, okay, so all of you P spread engine over here, it's not a difficult thing, it's not a difficult thing, can somebody tell me, it's a very simple question for all of you, can somebody tell me what is this angle, I want SD length somehow, this I know is R and I want SD length, can somebody tell me SD length and for that I need this angle, can somebody tell me what is this angle, or anyhow I mean it's not necessary that you give me by the same logic, okay one thing you would all agree that okay, let's say I don't need that angle, let me remove that, do you all agree that this angle would be A, do you all agree that this angle would be A, this is pukka no problem with that, okay and you also told me a little while ago that this angle was, this angle was, this whole angle was pi minus 2C, correct and this angle was A, so what is the remaining angle, so can I say this yellow guy is pi minus 2C minus A, ah big sigh of relief, okay that means this chotu guy that is your angle BSE will be pi minus 2C minus A which is actually A plus B plus C minus 2C minus A, why A plus B plus C because pi is the sum of the angles of a triangle, so I can write it as A plus B plus C, A gone, A gone, one of the C gone, so you'll get B minus C, so all of you get are convinced that this small angle over here is actually B minus C, okay now listen to my logic, this length is R, so this length SD would be R cos A, correct SD is R cos A, correct so in this triangle I'll redraw it again, in this triangle you know that this length is R cos A and you know this angle is B minus C, can you not find out the length of SE from here, what is SE length, what is SE length, you'll say it's a simple, it will be R cos A by cos B minus C, yes or no, yes or no, agreed, any questions here, any questions, please highlight, now my problem is solved, AS was R, SE was R cos A by cos B minus C, R R gone, so it'll become cos B minus C by cos A, okay but the problem is none in the answer or in the proof I don't have B minus C kind of a okay so what should I do in that case, okay don't worry I have a solution for that, what should I explain once again Kinshuk, SE length, SE, see this same thing over here I have just drawn for you over here, see this motion of my cursor, okay if this is R and this is A, SD is R cos A you are convinced with SD, SE, this angle is B minus C just now we figured out, just now we figured out B minus C, correct, so this angle is known, base is known, hypotenuse is what you want to find out now, you can find it out by your simple trigonometry, okay, now understood how SE length is found, okay cool, now all of you please pay attention, this is very critical over here, I mean I'm going to do some manipulations with this guy, I'm going to do some manipulation with this guy, so cos B minus C by cos A I'm going to multiply with 2 sin A, I'm going to multiply with 2 sin A, okay now sin A is as good as sin B plus C, isn't it, because you're dealing with a triangle, come on, yes or no, A is pi minus B plus C, so sin A is sin B plus C, go back to your trigonometry days, okay, now transformation formula, what is 2 sin B plus C cos B minus C, anybody will say it is sin 2B plus sin 2C, okay, and this is sin 2A, so what do I have, I have something very interesting for you, basically tells you in what ratio does S divide the join of A and B, so it means that S divides the join of A and E in the ratio, in the ratio sin 2B plus sin 2C, is it 2 sin 2A, I think this is going to finish up my problem, okay, so now let me go back to the figure again, so this guy S, this guy S divides this in the ratio of sin 2B plus sin 2C is 2 sin 2A, okay, and you already know X1, Y1 is the coordinate for A and E's coordinate is already known to you from here, this is your E coordinate, okay, so I'll make the diagram once again for you so that we can finally wrap this up, oh my God, three times we need to draw the diagram for this, yeah, so see this is a scenario currently for us, so let's say this was your S point, okay, this ratio is sin 2B plus sin 2C is 2 sin 2A, this guy was X1, Y1 and this guy was X2 sin 2B plus X3 sin 2C by sin 2B plus sin 2C, comma Y2 sin 2B, Y3 sin 2C by sin 2B plus sin 2C, correct, so what should be the coordinate of S, this was my A, this was my E, so what should be the coordinate of S, now I think we are all equipped enough to do that, so this multiplied with this guy, so I'm not writing the multiplication result, sorry I'm not doing the multiplication, I'm just writing the final outcome from it, so this will be giving you this term, okay, and sin 2A multiplied with X1, that is a simple looking term that is X1 sin 2A divided by the sum of these two, sum of these two ratios which is actually sin 2A sin 2B sin 2C, okay, can you just do the same thing for Y as well, can you do the same thing for Y as well, so Y will be Y2 comma Y2 Y2 sin 2B Y3 sin 2C plus Y1 sin 2A by, and I think this is what we wanted to prove and I think this is what we wanted to prove done, okay, big proof right, but I'm sure after you have understood how to get the coordinate of E and after you have understood AS is to SE, job is done, that you can carry out the remaining part of the question from your end also, is this fine, my God, the whole screen is full, okay, complicated, but I feel this is not beyond our understanding, it is definitely within our understanding, okay, can you go on to the next problem of a similar nature, sir, if similar nature then no need, similar nature, yes, or should I give it as a homework to you, what do you want, what do you want to do, do it now or you want to struggle at home, is better you do it now only, yeah, let's go, let's do it, Dave, bring it on, sir, bring it on, we'll solve it, okay, fourth point that we are going to talk about is orthocenter, what is orthocenter, my dear, orthocenter is the point where the altitudes meet, isn't it, so if I draw the altitudes, altitudes, it is not a perpendicular bisector, let me tell you, it's an altitude, that means this is 90 degree, that is, the foot is 90 degree, okay, so the meeting point of the altitude of a triangle is basically called its orthocenter, so let's say a, b, c, okay, orthocenter is represented by h, okay, now let us prove this as a question that the coordinates of the orthocenter is given by x1 tan a, x2 tan b, x3 tan c upon tan a plus tan b plus tan c comma y1 tan a, y2 tan b, y3 tan c by tan a plus tan b plus tan c, okay, so how do I do this question, let me again do the needful, let me again call this as d, so first get the coordinates of d, this is my request to you, let's get the coordinate of d, how will you get the coordinate of d? Approach is same, see similar approach, homework that we won't solve, yeah and now your semester exams will also begin now, so centre work will take a backseat, when is maths for R and R, tenth, tenth is, yeah a lot of time, a lot of time, chelo I'll also help you, can somebody tell me bd is to dc, sir if you could tell this, you could get the answer for d, no, okay, not to worry, I have a small simple question for you, what is this angle, okay let's talk about this length, what is this length, bd, if you see, if you see, this length is c, this length is b and this length is a, okay, this angle over here, if I'm not mistaken, that is actually 90 degree minus b, am I right, right, of course this is 90 and this whole thing is b, so what is this left, 90 minus b, correct, if this length is c, what is b, what is bd length then, you'll say bd length will now become c sin 90 minus b, that is c cos b, yes or no, correct, any doubt, any concerns here please highlight, what is dc length then, same logic, so this angle will be 90 minus c, no doubt about that and this length is b, so that is going to be b cos c, thank you Kinshok, clear, okay, now we all know from our sign rule that c is equal to 2r sin c and b is equal to 2r sin b, can I write it like this, 2r 2r gone for a t, sin c by cos c is tan c and sin b by cos c is tan b, so can I say, can I say, can I say my dear students this is a big breakthrough for us, the ratio bd is to dc, is tan c is to tan b, wow, that's a big breakthrough for us, correct, because your answer also has similar kind of a term, tan a tan b tan c, correct, so it's a big achievement for us, okay, so coordinate of d would be what, coordinate of d would be tan b x2 or you can say x2 tan b, x3 tan c by tan b plus tan c comma y2 tan b, and I dislike writing y2 because same thing you have to copy again, but what to do, you have to write, yeah, okay, alright, now let us focus on what is ah is to hd, what is ah is to hd, let me make a separate diagram for it, meanwhile note this down, this is a very important thing which we will use little later on, decoordinate we will use little later on in our final answer, okay, so note down d, note it down, let me make a fresh diagram, let me make a fresh diagram, fresh, okay, so this is my altitude, down, down, down, okay, a, b, c, hd, now my problem statement is what is ah is to hd, okay, for that let me find out h, ah ruchita, sign law, ayu, ayu Ramakrishna, sign law, should I write it, do you want me to write it, a by sign a is equal to b by sign b equal to c by sign c equal to 2r, thank you, so my question is what is ah, who will tell me ah, now all of you listen to this very carefully, this is 90 degree, correct, this length is c, correct, this is angle a, correct, okay, let me call this for the time being as af, so what is af, what is af, what is af, af, af, af, af, af, first what is af, somebody come on, this is a, sorry this is c length, this angle is a, what is af, this is 90 degree, af, tell me somebody, come on, c cos a, yes or no, yes or no, have I said something wrong here, af is c cos a, it's the right angle triangle, hypotenuse is c, this angle is a, what is base, base is c cos a, class you are there, some of you would have fainted also now, I use that, when will this okay, Ritu is the one who confirms it, okay, okay, what is h, see now this angle we all know is 90 minus c, little while ago only we figured it out, right, 90 minus c and your base, this is 90 degree, base is c cos a, so what is your hypotenuse, what is your hypotenuse, what is your hypotenuse, yeah, so you will say c cos a, no, hypotenuse will be smaller than the length, it is not possible now ever, so it will be c cos a divided by cos of 90 minus c which is sin c, is that fine, agreed, any questions, okay, good, so what is hd now, what is hd, now I can play the same game now in this way also, in order to know hd, okay, I already know this angle, this angle is 90 minus b and I already know this side, right, what is this side, what is this side, this side is b cos c, correct, so what is hd length, what is hd length, so hd by b cos c, so I will write it over here, hd by b cos c is tan of 90 minus b, correct, so hd is cot b into b cos c, oh god, sorry I was disconnected, can you all hear me, okay, can you all see my screen, so hd will be what, hd will be, so hd by b cos c is, is tan of this angle, tan of this angle is this, so I have written it as cot b, so I took the b cos c on the top and I also converted cot b as cos b by sin b, okay, now all of you please pay attention here, ah is known, hd is known and one very important thing, you can write it in a very simple term, c by sin c is actually 2r, okay, and this is equal to 2r cos b cos c, correct, so this is also 2r, this is also 2r, oh sorry, so ah by hd is equal to, now this question mark becomes, this question mark becomes cos a by cos b cos c, okay, now how do I solve this, because I need this in the ratio of tan, I need this in the ratio of tan, so what will you do in this case, now all of you please pay attention, I will multiply this with sin a on the top and sin a in the bottom, in the denominator, okay, I am moving very slowly over here in case you have any doubt, please feel free to stop me, okay, no doubt till this, this point of time, now sin a, I can write it, let me write it on top, sin a I can write it as sin b plus c, sin b plus c expansion you can write, I will write it in shortcut, sin b cos c and again multiply with a cos a plus cos b sin c, again multiply with a cos a, whole thing is divided by sin a cos b cos c, okay, if you literally bring the division, if you literally make the division happen, you would realize that when this term gets divided by this, you will get rid of what does it become, what does this become, is the expansion clear, I mean I hope you have written the expansion correctly, yeah, what does it become, okay, don't divide it like this then divide by, divide by cos a cos b cos c, divided by cos a cos b cos c, okay, if you do that, you will end up getting, you will end up getting tan a, check here, this will give you, sorry, tan b, this will give you a tan c and this will give you a tan a in the denominator, tan a in the denominator, okay, so basically it will become tan b plus tan c by tan a, by tan a, correct, so now what have you figured out, you figured out that a h is to hd, this is to this, is known to you right now, so let me make a third triangle, so let me make a third triangle over here, yeah, I will not write a lot of stuff over here, I will just write hd and a, so this is your x1, y1, this point was, correct me if I am wrong, x2 tan b, x3 tan c by tan b plus tan c, comma y2 tan b, y3 tan c by tan b plus tan c, okay, and this ratio was tan b plus tan c is to tan a, so what will be the coordinate of h, so anybody can now tell me the coordinate of h, so it will be this multiplied with your x coordinate, tan a multiplied with the x coordinate of a and then add the ratio, so overall if you see it will become, it will become, I just write first tan a into x1 which will make it as xa tan a and if this guy tan b plus c multiplies to this, tan b plus c will get cancelled off, okay, so leaving you with this, divided by some of these two, some of these two will be tan a plus tan b plus tan c, similarly y also you have to copy in the same way and this is our required formula, treat this as a learning, of course these coordinates may not be very helpful to you but treat this as how to deal with a geometry problem within a coordinate geometry, okay, so let's now move on to the next concept which is area of a polygon, area of a polygon when the coordinates are given, when the coordinates of the vertices are given, coordinates of the vertices are known, you would have all learnt your area of a triangle isn't it, how do you find the area of a triangle when the vertices coordinates are known, can somebody tell me what is the, I'll just take a, yeah, so let's say I have a triangle whose coordinates are known to me, okay, let's say x1, y1, x2, y2 and x3, y3, what are the area of the triangle you have learnt, right, so you have learnt that area of the triangle ABC in your class 10th, I'm sure you would have learnt this, it is half, it is half mod x1, y2 minus y3 plus x2, y3 minus y1 plus x3, y2 minus y1, correct, okay, is this the formula you have learnt in your class 10th, yes or no, 1, 2, 3, 2, 3, 1, 3, 1, oh sorry, 3, 1, 2, yeah, sorry, is this fine, there's a, there's a cyclic nature in this, 1, 2, 3, that's how I also remembered when I was a student, 1, 2, 3, 2, 3, 1, then 3, 1, 2, absolutely correct, okay, now how did you get this answer, did you ever try to question your teacher, how did you get this answer, did you ever try to ask your teacher how did you get this answer, no, no determinant, right, Dave, absolutely, so basically you dropped perpendiculars onto the x-axis, okay, like this and it created trapeziums for you, isn't it, so if you see there is two trapezium, in fact three trapezium created over here, let me name it as let's say ABC PQR, okay, so can I say area of the triangle ABC will be area of the trapezium BAQP, BAQP plus area of the trapezium ACRQ, okay, minus area of the trapezium which is down in the base over here, which is actually BCRP, yes or no, this is how you get it, correct, so how do you find the area of a trapezium, area of a trapezium is half into height into some of the parallel sites, isn't it, okay, so when you perform that you get this answer which I am not interested in getting right now, okay, you can easily confirm from here, okay, that this is going to be your answer, correct, now I will tell you some shortcut way to get this, okay, I am sure you would have learned this in your class 10th, Tushar sir would have taught you something which we call as the staircase method, staircase method or many books will call it as the shoelace method, okay, so this expression that you have, see when I was basically solving it I had, you know, by mistake written y2 minus y1 over here, so there was an element of confusion, okay, and it can happen in you also, it is very obvious right, our brain may not remember all the formulae, you know, throughout our entire preparation time, isn't it, hello, can you hear me, am I audible, can you hear me, hello, can you see my screen as well, can you see my screen, all of you can see my screen, sorry there was a power fluctuation from my end, I am extremely sorry about that, yes now there is a method which we call as the staircase method or the shoelace method which says you can easily find out the area of the triangle by using this simple trick, so write down the coordinates like this x1, y1, x2, y2, x3, y3 and then back to x1, y1, okay, so this method says area of the triangle is given by, just do this operation, the meaning of the symbol is you have to do this operation, multiply this with this, multiply this with this, multiply this with this, okay, so first write that down, so x1, y2, x2, y3, x3, y1, okay, then multiply these guys, these guys, okay and just write it in the same way as you are done for the yellow ones, so y, x2, y1, x3, y2 plus x1, y3 and subtract the two results, okay, you may have to take a mod in this case, okay, so you understood what I did here, I multiplied x1, y2, I wrote here, I multiplied x2, y3, I wrote here, I multiplied x3, y1, I wrote over here, then I multiplied x2, y1, I wrote over here, I multiplied x3, y2, okay and I wrote over here and I multiplied x1, y3 and I wrote over here, so the yellow ones minus the white ones would be your area of the triangle, okay, now this rule basically is nothing but it's an observation from this answer itself, okay, if you see this answer you have written x1, y2, see there is an x1, y2 here, you have written x2, y3, see there is an x2, y3 over here, you have written x3, y1, see there is an x3, y1 over here, then you have written minus x1, y3, minus x1, y3, if you open the bracket it becomes minus x1, y3, here is also minus x1, y3, see last term, minus x1, y3, then you have a minus x2, y1, then minus x2, y1 and then you have minus x3, y2 and then you have a minus x3, y2, okay, so I basically remember this staircase method and this always, this always helps me to get my answer right, okay, now one small thing I would like to highlight over here, if you take your coordinates of the triangle in such a way, if you take your coordinates of the triangle in such a way that while moving from x1, y1 to x2, y2 and x2, y3 to x3, y3 and from x3, y3 to x3, y1, so while moving from x1, y1 to x2, y2 while moving from x2, y2 to x3, y3 and from x3, y3 to x1, y1, your area is always to your left hand side, okay, let's say you are walking on this path, let's say you are walking on this path, okay, so when you are walking on this path, this area is always on your left hand side, okay, then your answer for this quantity will always come out to be positive, you don't have to take, you don't have to take a mod, now why does it happen? It is because when you take this, we basically follow something which we call as the right hand thumb rule, now see my right hand, okay, this is my right hand, you try to move along the arrow direction, so if you move along the arrow direction, you will realize that the thumb is coming towards your face, okay, when thumb is coming outside, it is a positive area, when thumb is going inside, it will be a negative area, let's say if I go from, if I go from this direction, let's say I go from B to A to C, so if I go in this direction, then what will happen? Try to curl your finger in the direction of the arrow, okay, naturally, you'll see that your thumb is going inside the plane of the laptop or the computer which you are using, in that case your answer would come out to be negative, okay, so when our teacher used to tell us in our junior classes that put a mod, she basically thought that you would take it in a random fashion and you can get both positive or negative of your value, that's why as a precautionary measure, she told you to take a mod, but in reality if you take your coordinates, let's say if you arrange them on the xy coordinate axis and you always take x1, y1, x2, y2, x3, y3 in such a way that while moving from A to B, B to C, C to A, your area is always towards your left hand side, that means you are following the right hand thumb rule, then there's no need to take a mod, your answer will automatically come out to be positive, are you getting my point, okay, so please note down this so that we can start applying it, okay, so now the topic name was area of a polygon, I just gave you area of a triangle, so can we use the same or can we generalize the same even for a n-sided polygon, if yes, how does this rule change and what is the shortcut for getting the answer, so the shortcut for getting the area of a polygon, so let me generalize it over here, so if you have or n-sided polygon, okay, I'm not drawing all the vertices, okay, let's say I put a dot, dot, dot, so if you have let's say x1, y1, x2, y2, dot, dot, dot and let's say this length, this was your xn, yn, okay, what is the area of this polygon, area of this polygon as per the shoelace formula will become half mod of or you can write mod outside, mod of this structure x1, y1, x2, y2, x3, y3 till you reach xn, yn and at the end you have to write x1, y1 again, okay, so this is something which is important, please do not forget at the last you're writing x1, y1 again, right, so this also will go like this, x1, y1, x2, y2, x3, y3 till xn, y1 and at the end you'll have x1, y1 again, then again do these multiplications which I told you, okay, like this, write them down and then do these other ones, okay, write them down, subtract both the results, take half factor of half and mod it if you're getting a negative answer, if your area is always taken in such a way that it is following the right hand thumb rule, that means if you're going from x1, y1, x2, y3 in such a way that your thumb is coming towards your face, then you'll always get a positive answer, don't worry about that, is this fine, so shoelace method is a very, very time efficient way to get our answer, later on we learn that there is something called determinant also which basically comes from the same similar idea, let's take a question, let's take a question, question is find the area of the quadrilateral ABCD whose vertices are given to you, okay, I'll try my best to plot the point as close as possible, let's say 1,1 is here, this is your A, 7,-3, 7,-3 will come down, so 7,-3, so B will come over here, C is 12,2, so 12,2 will still come up there, this is C and D is 7,21, 7,21 will be somewhere over it, okay, I'm just rough, okay, so this is 1,1, 1,1, this is 7,-3, 12,2, 7,21, this is my quadrilateral, yes, yes, Ananya, I'll show it again, this is the general formula, x1, y1, x2, y2, x3, y3, so on till x and yn and x1, y1 again, okay, now take this in such a way that you are always, you are always taking these points in such a way that as you move from x1, y1 to x2, y2, let's say I call this as x1, y1, so let me begin with this as x1, y1, this is x2, y2, this has x3, y3, and this has x4, y4, so when I'm going from A to B, B to C, C to D, D to A again, your area is always on the left side, so you'll see that you will not require a mod, you will not require a mod, very good Archit, Archit has got it, okay, so half, see here, those who have not followed the formula, very good Archit, so let's say I write 7, 21 first, okay, then 1, 1, okay, is better to write a lighter figure on the top because you'll repeat the same thing again, doesn't make much of a difference by the way, so 1, 1, then you write 7 minus 3, then you write 7 minus 3, then you write 12, 2, okay, then you write 7, 21, then you write 1, 1 again, okay, so please see how this is expanded, so first multiply the ones which I'm showing with the green side, this with this, this with this, this with this, this with this, okay, so let me write it down, half, yeah, so minus 3 plus 14, how much is 21 into 12, 21 into 12, this is 42, 12, 162, okay, and 7, this will come under 1, amrela minus, now multiply these guys, this with this, this with this, this with this, this with this, so this will become a 7 minus 36 plus 14 plus 21, okay, so this will give you, let's simplify this, I think 14 will get cancelled, even 7 will get cancelled, so 7, 14, 7, 14 will get cancelled, so you'll end up getting, this is 159 and this is plus 15, okay, how much is this, oh sorry, this is 21, this is 250, oh, 250, so this is 249, 249, thank you, so this gives you 264 by 2, 264 by 2 is 132 square units, okay, as you will see here, I have never taken a mod, this is what I was talking about, if you're translating on this figure in such a way that area is always, let's say you're walking, let's say this is a lawn, okay, let's say this is a, you know, garden on which you're, on whose roads you are walking, if your garden is always on your left side, as you are walking, then that area that will come out will automatically be positive, you don't have to make a separate mod thing for that, are you getting my point, okay, so 132 square units, I hope everybody has got it, no doubt, no concerns here, any questions, okay, let's move on to the next concept, next problem by the way, I'll give you a break after this, read this question carefully, after this question Pakul, I'll give you a break, P, Q and R points are as given to you, P is 2, 3, R is 4, minus 2, sorry, Q is 4, minus 2, R is alpha, zero, find the value of alpha if PR plus RQ is minimum, let's do the first question, all of you please first focus on the first part, sorry, we have to take directions into consideration here, why will you take a direction into consideration, oh you're talking about PR being a directed length, no, no, no, here it is just a length, okay, see there's no vector sign on top of it, Ruchita, okay, so you can take it as a length, okay, so I got an answer from Ruchita, we'll see Ruchita, P is here, Q is somewhere over here, okay, and R I believe is somewhere over here, very good Ananya, very good Ruchita, come on guys, Shambho, are you there, Amruta, Adarsh, Akshath, Aryan Jha, Aryan, Aryan Kumar, Gauri, Titeksha, yeah I was about to take your name Titeksha, okay, see guys, first part is very easy, okay, if you want your PR plus RQ to be minimal, right, commonson says that it can only happen when PR and Q are in a straight line, isn't it, okay, or anyways also if you see, if you apply triangle inequality, can I say by triangle inequality, in any triangle the sum of two sides is always greater than equal to, okay, the third side, so PR plus RQ will always be greater than PQ, correct, what does it mean, it means the minimum value of PR plus RQ will be equal to PQ, will be equal to PQ, correct, so PR plus RQ will be minimum when PR and Q are in the straight line that means R should actually be here, this is the actual position of R, okay, yes or no, commonson says that, correct, isn't it, so if PR and Q are in the straight line, if PR, Q are collinear, what does it imply, it implies that the area of the triangle PR, Q must be zero, okay, let's make a shoelace formula over here, so 2, 3, 4 minus 2, alpha 0, 2, 3, this should be zero, correct, yes or no, right, so this means what, half, okay, let's multiply these guys first, down, down, down, so this will be minus 4, 0, 3, alpha, now let's multiply these guys, 12 minus 2 alpha, 12 minus 2 alpha, this should be zero, correct, that means 3 alpha minus 4 minus 12 plus 2 alpha should be zero, that means 5 alpha should be 16, alpha is 16 by 5, there are few people who gave me this answer, let me check who were there, Ruchita gave me, very good Ruchita, I think before that Ananya gave, Ananya Raghunandan, awesome, very nice, this was not a very easy question actually, okay, good, any question, any concerns with this first part, all of you are happy with this, can we move on to the second part please, any questions please highlight, second part, find alpha if you want PR minus RQ mod, that is the difference of PR and RQ to be maximum, let's see, let's see who gets it, guys some of you have asked me will the classes happen during the term one exam, see if you have a gap on the next day, let's say today is Thursday and Friday there is no exam, okay then it will be a three and a half hour session as it happens normally, okay, but if there is an exam on the next day and there is a centre on the previous day then we'll keep the class duration as just two hours but without any break, will that be fine, any modification or any suggestion that you would like to give there, have you understood what I've said just now, if there is a gap on the next day, no exam on the next day then full three and a half hour session but if there is an exam on the next day whether it is for naffal people or for ajaji nagar people, the class will happen only for two hours, for use or you'd all be different right, I didn't get your, for us, whichever school has an exam will give the benefit of doubt to both the schools, don't worry, could you repeat the right hand number, yes see what I said was, I think that is not going to help you in this case Meghana, I'm just giving you the idea however, see if you want, if you want area of a triangle whose coordinates are known to you, let's say this is x1, y1, x2, y2, x3, y3, okay, let's say you arranged it in a xy coordinate system, okay and if you don't want to use mod, let's say I don't want to use mod then I should name it, I should take x1, y1, x2, y2, x3, y3 in such a way that as I'm moving from A to B, B to C, C to A, my area should always be on the left side of my hand, let's say I'm a person who's walking on this path, okay, so I'm walking so this is my right hand, this is my left hand, let's say I'm putting my hand like this, I'm walking on this thing, if the area always is on my left side, see the movement of my hand, then the answer doesn't require a mod, that is what I was trying to say, alternately if you curl your finger in the direction of the arrow, take your right hand, you can see the arrow being made on the triangle, just move your finger around that arrow, then see your thumb will always come out, if that is always coming out then you don't require a mod in your formula, it'll automatically be positive, very good Ananya, awesome, you cracked it, sorry, let me show you the question, question got hidden actually, yes, find the value of alpha if pr minus rq is maximum, yeah Akshath, I have got your message before also, I mean the last time you messaged me was was was was was, no I lost that because I got disconnected no in between, no previous question you didn't answer anything, at least to me, I got from Archit, Ananya, Ruchita, Titeksha, Madhumati, Ritu, Madhumati again, no I'm getting your answers, don't worry, okay good, Anusha, good, Titeksha, good, let's discuss it, I don't want to take away your break time, so let me redraw this figure once again, let me redraw the figure, Akshath, very good, now I got your answer, so let me redraw the figure once again, p2,3, q, what was q, 4,-2 if I'm not mistaken, 4,-2 and r is somewhere, I don't know where is r, r is somewhere, okay, so let's say r is here, okay, you want pr, you want pr-rq to be maximum, isn't it, pr-rq, pr-rq to be maximum, okay, now let us make the mirror image of q about the x-axis, so let's say this is the mirror image of q about the x-axis which is q dash, so q dash coordinate will be 4,2, right, oh yeah, there are 2 Akshath, who is the second Akshath, no, no, no, no, there is 2 Akshath d, let it be Akshath, we will figure out who is the person, I'm just tracking the IP of that person, he left the call, okay, yeah, so what I was saying, let's say, let's say q dash is the mirror image of q about the x-axis, now all of you please focus here, can I say if q is the mirror image, sorry, q dash is the mirror image of q about the x-axis, this length and this length would be the same, correct, so rq would be rq dash, okay, now, so if I have to find out the maximum of this, that means I have to find out the maximum of this guy, so what is the maximum of this guy, that is what we want to find out, correct, now why I did, why did I take a mirror image is for this reason that, now look at the triangle, look at the triangle p rq dash, okay, so focus on this triangle, can I say in any triangle, the difference of two sides will always be lesser than equal to the third side which is rq dash, oh sorry, pr minus rq will be pq dash, yes or no, yes or no, okay, now if you want your mod pr minus rq to be maximum, see it is always less than equal to this, so if you want it to be maximum, then pr minus rq dash mod should be equal to pq dash because pq dash is the maximum value that this can take, okay, now if somebody says pr minus rq is giving you pq dash, what does it mean in that case, it means that, it means that your pq dash and r they will become collinear, see what is the meaning of this term, if you are saying pr minus rq dash is pq dash, that is see, let me remove my camera, pr minus rq dash is equal to pq dash, it can only happen when prq dash or pq dash rr collinear, so this means pq dash and rr collinear, correct, yes or no, if they are collinear, can I say area of the triangle formed by pq dash r must be 0, correct, so half, let's use the shoelace method, what is the coordinate of p, p is 2 comma 3, what is the coordinate of q dash 4 comma 2, what is the coordinate of r alpha comma 0, so basically this is what I am going to use, correct, so let us expand it, so 4 0 3 alpha, so it's 4 plus 3 alpha minus 12 plus 2 alpha, this should be 0, in other words minus 8 plus alpha should be 0, alpha should be equal to 8, so this is going to be your answer, is that fine, is that fine, any questions, any concerns, clear, Dave, okay, so time for a break now, let's have a break, right now the time on my system is 6.32, but I'll give you a few gray seconds, 6.33 p.m., let's meet at 6.48 p.m., in short, I'm not looking at your messages, okay, thank you, let's meet at exactly 6.48, all right, I hope you people are back, back, okay, next concept that we are going to talk about is the concept of shifting of origin, this is the same concept that we had learned in our bridge course also when we talked about transformation of graphs, is my voice echoing, is my voice fine, can you all hear me properly, kind of, kind of, it's fine first engine, okay, fine, so shifting of origin is basically a concept where you try to shift your origin in such a way that you meet or you basically solve a given question with less effort, it's a concept which is very badly dealt with in school, in fact, in many of the schools they don't even talk about it, but without this concept, it will be very difficult to solve questions especially related to the concept of homogenization, now what is this homogenization, I'm using several terms over here which probably you have not heard of, you will hear it in a very quick time when we are doing more of coordinate geometry, especially when we are doing pair of straight lines, so what is this concept, what is the concept of shifting of origin, see, let us say if you are talking about a point 2, 3, or let's say I talk about a point 5, 6, okay, what are the meaning of this 5 and meaning of this 6, 5 basically represents what is the distance of a point from the y-axis and 6 represents what is the distance from the x-axis, correct, basically you have fixed your origin somewhere and with respect to this as your reference point, this coordinate is 5 by 6, okay, no doubt about it, let us say I decide to choose a different point as my origin, let's say I make another coordinate axis over here, now this is my new x-axis, let's say capital X and this is my new y-axis, let's say capital Y, okay, if I say, now my new origin instead of O has now become O dash, now this O dash is located at let's say 1, 3, what do you think would be the coordinates for the same point P, what do you think would be the new coordinates for the same point P, so will it remain 5, 6 or will it change, will it remain 5, 6 or will it change, of course it will change, right, how will it change, how will it change, so if you are basically bringing your origin to 1, 3 means you are shifting one to the origin one to the right and three up, right, that means you are coming one closer in the x-axis and three closer in the y-axis, so now the same point will now be 4, 3, absolutely, that will become 4, 3, correct, so remember when you are shifting your origin, the same position will now be known by a different coordinate, are you getting my point, so I have not displaced P anywhere, but since I've displaced my origin, my point P would be known by a different coordinate altogether, my point P will be known by a different coordinate altogether, okay, so shifting of origin will make coordinates change, it will make equations change, but remember it will not make dimensions change, are you getting my point, what are the meaning of dimension, for example let's say if I'm talking about a circle, if I shift the circle anywhere in the space, of course its equation will keep on changing, its center coordinates will keep on changing, but what will never change is the length of its radius, correct, so shifting of origin basically brings about a change in the coordinates, it will bring about a change in the equation, but it'll never bring about a change in the dimension of that particular figure unless and till the dimension is related to the origin, for example if you want to find out the distance of the center from the origin, probably that will change, okay, so whatever we did just now, can we generalize this, can we give it a more generic structure, so let us say I have, I have a point x comma y, small x comma y, okay, and let's say this is your origin, okay, now I decide to shift my, I decide to shift my origin in such a way that there is no rotation happening of any kind, that means my new x axis is still parallel to the old one, my new y axis is still parallel to the old one, so from this point 0 comma 0, I have brought it to this point, let's say o dash which is h comma k, so very same point p, what will be the new coordinates over here, so you'll say the new coordinate would be nothing but small x minus h and small y minus k, in other words your earlier x coordinate is the new x coordinate plus h and your earlier y coordinate is your new y coordinate plus k, this is something that you need to keep in your mind, okay, so if you shift your origin to h comma k, your small x will get replaced with capital X plus h and your small y will get replaced with capital Y plus k, are you getting this point, okay, now this is something which will help you in also changing the equation of a curve, so if you had a curve y equal to f of x, okay, when your origin was at 0 0, let's say if you shift your origin to h comma k, how would the equation of the curve get changed, so if you follow this your y will get replaced with y plus k and your x will get replaced with x plus h, something very similar to what we did in our bridge course, right, so now people ask me, so in bridge course when you're shifting something to h comma k, we actually replaced when the curve got shifted by h to the right and k to the up, we replace our x with x minus h and y with y minus k, right, basically you're doing the same thing over here but in from a different perspective, see if I have a parabola, I'll just take a simple example because parabola is something that you have done to a very large extent in your bridge course, okay, let's say vertex is at origin, okay, if I shift my parabola like this, such that the vertex now becomes, in fact the vertex now becomes let's say 2 comma 1, how would the equation of this parabola change, what would be the equation of this white parabola, we have shifted the curve two units to the right, one unit to the top, what would be the equation of the parabola, no practical, let's not write, by the way this is x square is equal to 4y, my bad, yeah, x square is equal to 4y, yeah, how would it change, tell me, so you'll replace your x with x minus 2 and y will get replaced with y minus 1, correct, this is what you had learned, correct, now actually what is happening here which I didn't tell you in the bridge course because you would not be able to appreciate it at that point of time, your curve is not shifting, your origin is shifting actually, so this yellow curve remained wherever it is, it's just that I brought the origin two units to left and one unit down, are you getting my point, so in this case your origin got shifted to minus 2 comma minus 1, guys it is not the curve which is shifting, see curve will remain as it is, origin got shifted from here to here, are you getting my point, I think my picture is the mirror image of what actually I wanted to do, let's say the origin is here, let's say origin is here, this is my origin over here, the curve shifted actually, origin shifted over here, are you getting my point, so if your origin got shifted to h comma k, this is your h and this is your k then what do we normally do, we replace our x with x plus h and we replace our y with y plus k and that is what you actually told me, so both the answers are actually the same but just seen from a different point of view, you see it as a shift of the curve, I see it as a shift of the origin and that time also if you go and watch those videos back, you would see many a times I have used this word, it is not the curve which is moving, it is actually the origin which is moving, okay are you getting my point, so let's say I want to know the distance, if I want to increase the distance of wall from me, obviously I cannot move the wall, I have to move backward, no, no, no, no, no if the curve has to move top and right, origin has to move down and left, getting my point, so think that your white curve is actually your yellow curve only and origin got shifted, yeah I'll repeat once again, see earlier your diagram was like this, right, correct now let's say the curve is here only, I shift my origin to this position, getting my point, so isn't this now the white diagram that you have seen over here, so this is what I'm connecting to this guy and this is what I'm connecting to this guy, ah got it now, so curve doesn't shift, I'm repeating it again, curve doesn't shift, so in bridge course try to recall, bridge course what did I say, when you shift the curve right by h unit, you replace your x with x minus h, correct remember, but it is in reality the curve is not shifting h units to the right, origin is shifting h units to the left, correct, so origin is going to minus h comma zero, so your capital x as you can see from the formula your capital x will be small x, your small x will be capital x plus h, so your new coordinate will be capital x, so it will be capital x minus h will come there because you're shifting to the left, okay, anyways I have not written any sign to h and k over here because h and k can be positive negative any of the quantity, just keep this formula in your mind, small x is capital x plus h, small y is capital y plus k, nothing else you need to remember, nothing else you need to remember, okay, so with this formula you can find anything you want, you can find the new equations to the curve, see when coordinates axes shift, when your origin shifts, your equations will undergo a change, how will that change that is decided by this formula, okay, let's take few questions to understand this, guys get this right else there will always be this sign confusion, there will always be a sign confusion if you don't get this right today, question is at what point should the origin be shifted if you want 4 comma 5 to become minus 3 comma 9, so see this is your x comma y, this is your capital x capital y and just remember that relation capital x is small x or you can say small x is capital x plus h, let me write it in the reverse way, capital x plus h is your small x and capital y plus k is your small y, this is your formula which you need to remember in your mind, okay, so 4 is equal to minus 3 plus h and 5 is equal to 9 plus k, okay, so your h will become a 7, your k will become a minus 4, answer would be your origin must be shifted, your origin must be shifted to, shifted to 7 comma minus 4, 7 comma minus 4, absolutely correct in shape, is this clear, is this clear, any questions here, okay, just remember small x is capital x plus h, small y is capital y plus k, this is the formula that you need to remember for your shift of origin, let's take another question, now this all cases that I'm telling is when your origin has, oh sorry, when your axes have not been rotated, okay, you are not rotating your axes, I will talk about rotation also in some time, okay, so let's do this question, if the origin is shifted to 1 comma minus 2 without rotation of axes, what do the following equations become, let us first do this one, everybody please try it for one and a half minutes, then we'll discuss it, done first one, nothing, nothing, why will you complete this square here, what is the formula which I told, small x is capital x plus h, small y is capital y plus k, that's it, this is what is, okay, okay, so now you have to replace your small x with capital x plus h, what is h here, what is h, h is 1, you have to replace your small y with capital y plus k, k is what, minus 2, getting it, so when you do that change in the first one, you get 2 x plus 1 the whole square plus y minus 2 the whole square, minus 4 x plus 1 plus 4 y minus 2, simple, you are simplified, so if you simplify it, you get 2 x square, you will get minus 2 x, okay, in fact let me write y square terms first before I write anything else, you will get y square, this will give you 4 x, 4 x will get cancelled off, minus 4 y and 4 y, that will also get cancelled off and you will be left with 2 here, 4 here, minus 4 here, minus 8 here, oh, this will give you 2 x square plus y square is equal to 6, is that fine, did anybody get that answer, everybody got this answer, anybody who got a different answer, please let me know, clear, okay, try the second question, similar way, second question, y square minus 4 x plus 4 y plus 8, so what will you do, y minus 2 square minus 4 x plus 1 plus 4 y minus 2 plus 8, okay, let's expand, so y square minus 4 y plus 4 y will get cancelled off, then you will get minus 4 x, okay, and 4 here and 4 here will get cancelled off, 8 here and 8 here will get cancelled off, so it will just give you y square minus 4 x equal to 0, is this fine, does it make sense, clear, okay, we'll do more questions, we'll do more questions, don't worry, enough questions we'll take, okay, find the equation to which this equation is transformed, if the origin is shifted to 2 comma minus 3, the axis remaining parallel to the original axis, that means there is no rotation happening, guys, if rotation comes into picture, the formula will change completely, which I will deal in some time, okay, let's not simplify it, let's not waste time simplifying it, just give me the raw form of the expression without simplification, okay, I think it will be too much for you to type also, what should I replace x with and what should I replace y with, at least tell me that, what should I replace x with, write like this, x plus minus what and y, sorry, y should be replaced with, right, x should be replaced with capital X plus 2, y should be replaced with y minus 3, absolutely, so when you simplify it, you have to write your result like this, of course I don't want you to waste time doing it, it's a futile effort, once you have understood this, there is no point wasting time doing it, okay, minus 26, y minus 3 minus 60 equal to 0, okay, one small thing I would like to add over here, which I forgot to add here, normally we don't leave our answer in terms of capital X and capital Y, okay, have you ever seen any equation written in capital X and capital Y, right, so what we do is, we gently convert x to small x and y to small y, okay, so capital X and capital Y was only written for you to make a distinction between the old equation and the new equation, correct, but when we are representing our final answer to the examiner, we have to use small alphabets only, got the point, so don't, don't write things in capital X and capital Y, okay, it is not a norm to write any equation in terms of capital X and capital Y, okay, let's take one more question, so there was a curve, the equation of a curve referred to the new axis, new axis, yeah, referred to the new axis retaining their direction and origin being 4 comma 5 is given by this, now they have written it in a capital letter just to tell you that this is the new equation, this is the new set of, this is the new equation for you, now they're asking you what was the original equation, what was the equation referred to the original axis, you got the question, no, so there was some curve, there you replace your x with capital X plus 4 and y with capital Y plus 5 and you got this answer, so what was the old curve that they're asking you, absolutely very good, correct, so see small x is capital X plus H and small y is capital Y plus K means capital X is small x minus H and capital Y is small y minus K, correct, so you replace your, you replace your capital X with small x minus 4 and capital Y with small y minus 5, that's it, story's over, so the answer is X minus 4 the whole square, Y minus 5 the whole square is 36, simple as that, clear, any questions, any concerns with shifting of origin, so this is the formula, only formula which you need to remember and all your kastas will be taken care of, okay, good, next concept in the next 17 minutes I'm going to talk about a very, very important concept, rotation of axes, okay, so as we all know that x y coordinates basically depends upon its distance from the reference axis which we call as the x and y axis, right, so any point p basically we refer its coordinate by stating its distance from the x axis and the y axis, okay, now of course shifting of origin I will change it but not only that if you rotate it that will also change your coordinates of a point, so let's say I have rotated my coordinate axes by an angle of theta degree anticlockwise, so let's say I rotated my coordinate axes to new position capital X and capital Y by rotating it by an angle of theta, that means this is also theta, won't the coordinate of the same point p change, won't it become some different coordinate, yes, yes or no, why because of course earlier this was your y and this was your x, correct, this was your x and this was your y, now this is your y and this is your x, correct, of course they will change and hence the coordinate will also change, so how will it change is something that we are going to now figure out in the next 15 minutes, okay, so now pay attention over here, this is a very important concept, very very important concept, I will remove all unnecessary things over here, okay, whatever I don't need, for this I will take help of the polar coordinates, the one which was given by Newton, okay, so let's say let's say this point p, let me connect it with the origin, okay, let's say I call this distance as r, so this distance is r, okay and let's say I call this angle as 5, okay, now Newton said that you can locate a point by referring to its distance from the pole which is the origin right now in your case and by giving this angle which is phi in this case and how is this polar coordinate connected to the Cartesian coordinate, we know that x is r cos phi and y is r sin phi, I think we have been learning this since trigonometry days not a surprise to us, correct, yes or no, now let us say I rotate my coordinate axes theta degree anti-clockwise, correct, so now think as if your x and the y axes have become these white lines as you can see, now what is this angle, this shorter angle, what is this angle, you'll say sir obviously it will be phi minus theta, correct, yes or no, correct, does it change the distance from the pole by rotation, rotation does it change the distance from the pole, does rotating coordinate axes change the distance of the point from the pole, no, it will still remain r then this distance will still remain r, so can I say by the very same logic by which we connected x, r and phi and y, r and phi, I can say capital X will be r cos phi minus theta and capital Y will be r sin phi minus theta, correct, yes or no, is there any doubt regarding these two then please talk me, clear, no problem, no concerns, okay, now let us work on these terms, I will start expanding it, so if you start expanding it this will give you r cos phi cos theta plus r sin phi sin theta, correct, so I just expanded this guy, so expand the other term also it will become r sin phi cos theta minus r cos phi sin theta, correct, don't worry those terms will come up, those terms will come up, don't worry about it, those terms will automatically come up, okay, now all of you please pay attention, all of you please pay attention, what is this guy, this guy is small x, see here, see here, r cos phi is small x, right, what is this guy, this guy is small y, what is this guy, again small y, what is this guy, again small x, correct, so what did we learn from this exercise, we learned that the new x coordinate will be old x coordinate cos theta plus old y coordinate sin theta and new y coordinate will be old y coordinate or you can write this first, negative of old x coordinate sin theta plus the old y coordinate cos theta, okay, more importantly, which I am giving up to you, leaving up to you for your homework, your small x is actually capital x cos theta minus y sin theta and your old y coordinate was your new x coordinate sin theta plus new y coordinate cos theta, this formula is what I want you to remember, this is more important than the previous one, this is more important than this one, why this is more important is because most of the questions will come asking you what is the new equation, okay, so new equation of curves if you want to find out, this formula will be useful, if you want to find new coordinates then this formula will be useful, so this is important if you want to find out the new equation, which is more commonly asked, and this is important when you want to find out the new coordinates, now how is this transition from here to here, that is something which I am leaving up for you to do as a homework question, okay, sir in the box form the first equation was small y, yes my dear it was small y, I have just shifted it, I have changed the position, see it carefully, this was y, you know y cos theta I have kept it over here, this was minus x sin theta I have kept it over here, I just swapped the positions, yes I have a doubt, yes tell me, sir in the box formula that you have put for the new equation, yes the x, you have written x is equal to capital x cos theta minus small y sin theta, yes small y sir, yeah no no no sorry sorry, capital y, all of them are capital y, oh thank you, thank you for bringing it to my notice, yeah sorry, but the sign will change, this minus sign will actually come over here, this plus sign will come over here, so these two signs will interchange their positions, okay anyways I am leaving up to you to prove it, it is very easy, you just have to multiply that you know equation with cos theta, this with sin theta and subtract it you will get small x, okay similarly multiply this with sin theta, this with cos theta and add it you will get small y, okay that you can do for homework, now let us understand this formula through problem practice, let us take few problems, we still have 8 minutes left, we can take at least one problem in this time limit, let us take the question, question is the equation of the curve referred to the given system of axes is as shown on your screen, okay find the equation if the axes are rotated by 45 degrees the origin remaining unchanged, now guys one small thing I would like to bring it to your notice before you solve this question, we will go back, this formula was derived when your coordinate axes were rotated theta degree anticlockwise, does it mean I will have a different formula if it was rotated clockwise, no if it is rotated anticlockwise theta is positive, okay so theta is positive for anticlockwise rotation, so the question just says 45 degree means it is by default anticlockwise, okay and you will take theta as negative, I will write it down here with yellow, negative if you are rotating it clockwise, okay so formula is not going to change, it is the same formula that you are going to follow is just that when it is clockwise, when your axes are rotated clockwise you will keep theta as positive value, that's a plus 45 degree, had the same question mean if you rotated 45 degree, sorry anticlockwise positive clockwise negative, okay so in this case the problem that we have over here they have just written 45 degrees, 45 degrees means clockwise for anticlockwise 45 degrees, so this means anticlockwise 45 degrees, okay so positive angle means anticlockwise negative angle to be chosen for clockwise, okay so let us recall the formula your small x is capital X cos theta plus plus capital Y, let us look at the formula here whatever we have got capital X cos theta minus capital Y sin theta minus capital Y sin theta, so let me go back, yeah so minus capital Y sin theta and small y will be capital X sin theta plus capital Y cos theta, okay so very good Ritu, let us check, let us check, okay so your x will get replaced with capital X cos 45 which is 1 by root 2 minus sin 45 which is minus y by root 2, your small y will get replaced with capital X by root 2 plus capital Y by root 2, okay so just place it over here, so 3 capital X by root 2 minus capital Y by root 2 whole square plus 2 capital X by root 2 minus capital Y by root 2 times capital X by root 2 plus capital Y by root 2 plus 3 capital X by root 2 plus capital Y by root 2 whole square is equal to 10, lot of things we can do with these root 2 first of all they both have root 2 so you can bring it out, so it will be 3 by 2 X minus Y the whole square and root 2 root 2 will get cancelled so this will give you X square minus Y square, this will also give you 3 by 4 X square sorry X plus Y the whole square okay equal to 10, okay if you mix these two it will become yeah it will become 3 by 2 into 2 times X square 2 times Y square okay this will be X square minus Y square only equal to 10, let us simplify this further, so this is going to be this is going to be 3 X square 3 Y square plus X square minus 2 Y square equal to 10 which is 4 X square plus 2 Y square equal to 10 or you can say 2 X square sorry this is going to be yeah I am so sorry yeah this is going to be 2 X square plus Y square equal to 5, did anybody get this answer 2 X square plus Y square equal to 5 got it is the idea clear okay so through this entire exercise we learned that when we are shifting the origin let me summarize it over here I will not take up any question here I will just summarize it summary of shifting of origin summary of shifting of origin was your small X is replaced with capital X plus H and small Y is replaced with capital Y plus K okay and what is the summary of rotation of axes rotation of axes so remember your small X is replaced with capital X cos theta minus capital Y sin theta correct and small Y is replaced with capital X sin theta plus capital Y cos theta don't forget theta is to be taken as positive if rotation is anti clockwise and theta is to be taken as negative if your coordinate axes were rotated clockwise that's it this is the moral of the entire discussion that we had after the break got the point okay with this you can solve the dpp's that we have I will be sending you today