 Okay, so again, I'm repeating it. What is the parabola parabola is defined as the locus of a point Which moves in a plane so it must move in a plane in such a way that it's distanced from a fixed point And what is that fixed point called? focus Okay Now, I'm sure you're listening to this particular point for the first time in a circle the focus and the center both were Coincident and that was called center But for the first time you realize that there's something called focus as well So from the fixed point is equal to its distance from a fixed line and what is that fixed line called? The fixed line is called the direct tricks. Okay, don't start calling it as dielectric There was one interview which I was conducting long ago and what teacher was from Andhra Pradesh So I told him please explain parabola locus definition to me So he kept on saying dielectric dielectric dielectric and I was like what he's saying I mean, I started doubting my you know a knowledge about the topic So later on I corrected him. It's not direct dielectric. It is direct tricks Okay, so dielectric is something very different by which you put between the capacitors I'm sure you would have done capacitance in physics. Yes or no Yes So if you draw the scenario How would it look like? Let's say this fixed line is this red line and the fixed point is this fixed point s Okay, so let me say this line is L equal to zero line. Okay, and there is a point Let me call this as P point and this point is moving in such a way that its distance from This point is equal to the distance from this line when I say distance, I mean perpendicular distance. Okay So what I'm trying to say is This distance and this distance should be always be equal as it is moving Getting my point so SP should be equal to PM by the way Later on will you know come to learn this that this ratio That is the distance from the point divided by the distance on the line is called a century city Which we denote by the alphabet E. So he stands for a Centricity Okay, so in case of a parabola, what is the eccentricity? One one so SP by PM is equal to one That means your eccentricity is equal to one in a parabola. By the way, there was eccentricity for a circle also for a circle What was the eccentricity? Zero zero. Yes, please remember these following things for circle E is zero for an ellipse Which we are going to take up next after parabola E is between zero and one Let me write it properly. Yeah, E is between zero and one for a parabola E is equal to one for a hyperbola E is greater than one for a pair of straight lines E is tending to infinity Getting this point. Okay So now If this point now follows this geometrical condition and starts moving It is very obvious that the path that it will likely to be traced is this path. Let me write it more Let me draw it more accurately So somewhat like this Pass through this guy So somewhat like this Okay, and this path that you see is the the shape of a parabola. So it'll go on indefinitely Is that fine with the other good morning? Okay Just to show you a typical shape of how a parabola will look like on a geogibra tool So let me show you on a geogibra tool. This was a diagram. I was making yesterday Y square is equal to let's say Something like 4x. Do you see the structure? Okay, this is the typical structure of a parabola. Okay Now a few things which we need to know about the terminologies associated with it If you draw a line passing through the focus and Perpendicular to the directrix something like this This line is called the axis of the parabola So you can say axis is a line which No divides parabola into two symmetrical halves So, please be careful about the nomenclature which I'm telling you now because I'll be using it For problem-solving also. So what is an axis of a parabola? It's a line which is perpendicular to the directrix and Passes through the focus Okay Symmetrically dividing the parabola into two equal halves The point where the axis meets the parabola is called the vertex of the parabola. This is called the vertex Okay, if you connect any two points on the parabola, let me just take a different color over here Look if I just draw a line No connecting any two points on the parabola that line will be called as the chord of the parabola. Is that fine? if you draw a line which is passing through the focus Let's say I draw a line like this passing through the focus and connecting two points Then this pink line would be called as the focal chord So chord passing through focus will be called what focal chord Okay if you draw a line Which is always or if you draw a chord you can say which is always Perpendicular to your axis That means if I draw a line like this need not be passing through focus. Okay, then this line is called double ordinate double ordinate Okay, so what all we learned we learned this point is called focus This line is called directrix Please do not use dielectric for it This point is called the vertex Any line connecting two points on the parabola is called a chord If the chord passes through the focus it will be called as the focal chord If a line is perpendicular to the axis it will be called as a double ordinate so far so good Now one more thing is left off a line which is Let me choose orange Align, which is perpendicular to the axis and passes through the focus That means it is a double ordinate only but now it is passing through the focus Then that line is called as the lattice rectum Right, it means shortest line. Why it is called the shortest line is because Lattice rectum is the shortest focal chord. We should be proving it later on. So I'll write it here for your reference lattice rectum is Your shortest focal chord You cannot draw a focal chord You cannot draw a focal chord shorter than the lattice rectum Guys, is it clear? Yes or no? Is this part everybody? Yes, sir. Okay So I'm sure you are clear with the terminologies that we have associated with the parabola axis focus Directrix chord focal chord double ordinate lattice rectum. No doubt about any one of them Now the time has come that we derive the equation of a parabola Okay. I'm a how you so the time has come that we derive the equation of a parabola so for deriving the equation we need to first give Coordinates to certain points. We need to decide which is our origin We need to decide what is going to be my x-axis my y-axis because without fixing those reference points and lines We cannot write any equation. Okay so My heck I would request you to just take a snapshot of this page and don't worry I'll be sharing this PDF with you at the end of the class That's why it's very important to join in on time because the initial five ten minutes the agenda is set and If you miss out on that you have lost the touch of the topic So what I'll do is I'll redraw the entire thing once again Okay, and this time what I will do I will take a snapshot of it and keep because I would keep drawing it again and again So let me do a small Take over here Let me draw a circle Like this Okay, so what I'll do now I will erase half the circle Somewhat like this. I'll erase Almost looks like a parabola, right? So I'll just keep this as my image so that I can keep referring to it. Okay Now What I'll do now, by the way, this is not the direct tricks direct tricks is a separate line I'll draw it again separately. This is your direct tricks. Okay So yellow line is your direct tricks What I've done is I have taken the vertex to be the origin So vertex I have put it at as origin and I have taken the axis of the parabola to be your x-axis. I have taken a Line perpendicular to the axis and passing through the vertex to be your y-axis In other words, I have treated my y-axis to be the tangent at the vertex So basically this line is also the tangent at the vertex as You can directly see from the figure as a couple of things that I would like you to understand over here Let's say this is the focus Okay, normally we call it as s Do you realize that the vertex V is the midpoint between s and this value n? Do you realize that that means nv is equal to vs? That is to say V is the midpoint of V is the midpoint of n and s Okay So then once I fix my origin at V what I'll do is I'll take my s to be a comma zero Where a is this basically this distance Okay, now remember one thing guys, which I'll be you know telling you again and again in this chapter a is basically the distance between the vertex Distance between the vertex and the focus Hence a can never be negative. So a will always be a positive quantity. Okay Now if this is s is a comma zero, who will tell me what is n? Minus a comma zero minus a comma zero. So can I say the equation of this line is x is equal to minus a Correct Now what are the locus definition the locus definition of an ellipse is if you take any point on the ellipse Let me call h comma k It's distance from s should be the same as its distance from this line Okay, so let me call it as p. So sp should be equal to p. M. That is what we learned as a locus definition Okay Niyati May I know the reason why you are late by almost 12 minutes Sorry, sir. I just came now. Just came now means what? I Had gone out for breakfast Were you not aware that the class is at 9 o'clock? You saw it right now very good. I can't speak anything after that very good Same with with the other Guys, I'm telling you again and again in an online class. Please respect the time When the class starts and you lose out on the first few minutes everything is like, you know, go on Okay, so SP SP would be nothing but distance between H comma k and a comma zero. What will be that? Can I say it will be under root of h minus a the whole square plus k square? What is PM now? Can I say this distance is H? This distance is a So PM distance will be H plus a and just take a mod of it in order to ensure everything is positive Okay So this is basically Nothing, but you are establishing a local solution or you're using the local solution and Through that you are trying to get the relationship between H and K Hope you all recall that when you are finding the equation of the path taste by any point You are essentially finding what you're finding the relationship between the H and the coordinates of the point, isn't it? Okay, so what I'll do is I'll simplify this by squaring both the sides So H minus a whole square plus k square is equal to H plus a the whole square if you open this up You'll end up getting At square plus a square minus 2 a H plus k square is equal to H square Plus a square plus 2 a H At square at square gone a square a square gone So you end up getting 4 a H is equal to k square. Okay now you generalize here Now you generalize here on generalizing it we end up getting Y square remember k has to be replaced with y and H has to be replaced with x and This is what basically is nothing but the standard form Watch out for my word standard form of a parabola That means not all the parabola in the world look like this This is one of the very special cases which we call as the standard form Basically this form would be our This form is like the frog in our you know bio laboratory will do all the experiments on it That's fine because it's very simple to you know, right and express But let me tell you not all the parabola in the world would look like this There's just a standard form by the way There are three standard forms which I will talk about later on but before that one question is coming your way Any questions any concerns here? So one thing that is very obvious from here is that in a parabola In a parabola the vertex is always the midpoint of n and s Secondly axis is always perpendicular to the directrix and the locus definition is The distance of any point on the parabola any point you can take any point it can be a viewpoint also It can be this point also it can be this point any point you take its distance from s should be distance from this line This line this line. Okay, this line is called the direct X line Where do you see parabola in your, you know, practical life? Have you ever tried to question it? Where have you seen parabola in your practical or physical world? Said traveling hairpin bends Hairpin bends, okay Have you have you seen your car headlight if you see your car from the side It'll be like especially, you know, if you have a Toyota car and let's say they Expose the mirror from side. Those are parabolic reflectors. Okay Now there's a speciality of a parabola that any source of light at the focus After hitting the surface will become parallel to the axis principle. Yes principle axis So this is basically used to, you know, project the beam of the car parallel to the road Especially when you want to see far Okay, the same principle is used in medical world where let's say Yeah, in order to kill kidney stones, okay, so last year Basically, Manav had this kidney stone. So I took him to the doctor. So what the doctor did was he Kept his kidney stone at the focus and send some laser beams Okay, and after hitting the surface it passes through the stone and thereby destroying the stone So that's how Manav is there with us Manav, are you there? Yes, sir So this has a lot of application in the medical world also Okay, and apart from, you know, arches and all by the way, don't confuse a catenary with a parabola What is a catenary by the way? Catenary is if you take a string of uniform mass and you know tight between two points It normally takes a shape like this, isn't it? Like just like a necklace in your neck, isn't it? That shape is not a parabola by the way It's not a parabola. It's actually something which we call as a catenary catenary Catenary is the cos hyperbolic function. It is cos, you can say Okay, I'll show you the difference. I'm sure I'll show you the difference during the bridge course If you if you guys remember Just see over here, okay Y is equal to X square is a parabola as you can see on your screen Okay But Y is equal to cos Hyperbolic X Almost very similar. You can see that almost the blue curve is your cos Okay, but they're not the same things As you can see, they're not the same things Okay, so one looks like the difference is like the X square X4 function Sorry, if I shift it down Sorry, what are you saying? No, sir. One is Fatter one is thin so it looks like that difference between X square X4 functional. Yeah So basically they're not the same. That's what I wanted to so the blue one is basically the you know The structure which will be taken by that string or mark a rope of uniform mass, but that is not a parabola What makes it different sir See both are different type of functions This is cos cos is basically e to the power X plus e to the power minus X by 2. It's made up of exponential functions Whereas parabola is made up of polynomial functions, so they're not same Of course, they try to mimic each other, but they're not the same Okay, I'm not happy sir. Not happy. So watch a movie you'll be happy See there are many functions you come across which start mimicking mimicking mimicking mimicking each other for example X cube and tan X Yeah Both of them will be almost of this nature Yeah But they're not the same it's like the difference between ape and a man Okay, of course have the similar qualities, but you can't call it call them as the same It's the same between an orange and a musami There's a difference between them. All right musami is different orange is different right? Oh Okay, I Think it all is a continuous piece of mass. Okay, and I know it is a split There's a split happening. I think you're a bios friend. You should be doing it better Okay, sir Chalo next thing that I would like you to do. Yes, I told you you'll be doing a problem question. Yeah question So the question is there is a parabola whose focus is at negative one comma negative two Okay, and directrix has the equation X minus two y plus three equal to zero Okay, tell me what is the equation of the parabola? I'll give you two minutes for this take two and a half minutes only I just no need to type it out if you have solved it Just speak out the answer that's fine. Yes any success No, sir. It's a very very simple. See let me draw this figure in Actual sense so minus one minus two is a point in the third quadrant, right and This line is basically a line which is going to be like this correct me Yeah Possibly stop line. Sure. Okay. So this is your directrix and this is your focus So basically your parabola would be all Hope you're able to see that dotted line So that's your parabola. You have to find the parabola Now always remember any parabola will follow the definition that it's distance from the focus Should be its distance from the directrix Correct, so SP should be equal to PM correct So take the point P to be H comma K Okay, many people take X comma Y also that depends upon your convenience your comfort level So H comma K distance from S that is SP should be equal to PM So what are the distance of H comma K from minus one comma minus two under root of H plus one the whole square K plus two the whole square. Basically. I'm following the locus definition Guys at the end of the day, all the conic sections are locus That's why I keep on saying this Repetitively that you should be good in locus For J coordinate geometry means locus. Nothing else What is PM? PM will be more Okay, plus three by under root that that's good. Very good. Very good Hey mistake, you're doing over and over again when cut. Oh, yeah S square plus K square or one square plus minus two square Minus one square plus minus two square. Oh, when will you improve my dear? Okay, now cross multiply and square it When you cross multiply and square it you'll end up getting something like this Now you may square it open it and try to simplify it I'm doing it in one go. So it's square will be there. So five at square and here also a square will be there So they will come up to become four at square. Just keep correcting me if I go round somewhere, okay? Five K square from here and four K square from here will give you a K square HX term will be only coming from here So minus four HX come on this side will become plus four. Sorry, HK not HX by the way mistake. Yeah, HK What will be H term? So from here, I will get 10 H and from here So I'll get four K will be 20 K from here and from here. I will get minus 12. So that will make Constance will be five into five twenty five from here and nine from here. So that will be Okay, now you generalize this When you generalize this you end up getting four X square plus Y square plus four X Y Four X plus 32 Y plus 16 equal to zero as you can see It's like I'll start coming your equation can become as ugly as this So it is not always Y square is equal to four AX type. Okay? You can have see all the phonics I have told you in the beginning of the phonics chapter also They have a structure like this. So you can't you know The presence of other terms It can be going where X Y term can also be there, you know, both X square Y square term can also be there Yeah, yes, sir Why I'm telling you this is because you know after studying, you know NCRT people think the only equation possible Very very wrong that is a very special case that is One more important thing I would like to highlight over here since I've taken this problem the presence of X Y term Actually is a signifier that The parabola is oblique in nature oblique means slightly rotated Are you getting my point if your parabola was You know like this or like this or like this or like this Or a shifted cases of this that means without rotation then X Y term will never make appearance The presence of an X Y term is an indicator that there is a tilting happening in the axis of the parabola Right just for by experience. I'm telling you this. So please make a note of it. Is that fine everyone? Any questions here how to get the equation? So please do not forget the locus definition Yes, sir So one more question. Yeah, sure. I'll give you one more Find the equation of the parabola whose verdict is 3 comma 2 and focus is at 5 comma 2 Find the equation of the parabola One second. Yeah So y square minus plus 4 y minus 48 equals 0 plus 4 y minus 48 equals 0 Yeah Okay, there's no 48 Okay Anybody else who would like to answer you can type it out privately to me also. It's not a very big equation So you you can assume the y axis to be the direct x, right? Oh No, no, I got it. I got it one second, sir. Unless until it is there Yeah, I got I got my mistake one second there Two minutes actually not one second sir x equal to one line as the direct x x is equal to Is a direct x. Yes, you can say that. Ah, yes, that was my mistake One second sir Niyati, Vidyotha, what are you all are solving? Sir, he's got 43 this time Yeah, always in coordinate geometry the first thing you should do religiously is draw a sketch Sketch, I'm not saying graph sketch. Sketch is a rough estimation So 3 comma 2 again, so this is your vertex 5 comma 2 is here. This is your focus, correct So I'm sure the parabola would be of this Correct Now in order to know the equation of a parabola two things are required. One is focus. How are we known to us? Directrix How will I find the directrix very simple Directrix is the line which is perpendicular to the line connecting the vertex and the focus By the way, this line is parallel to x axis. So your vertex will be parallel to y axis y axis. Yeah Now secondly the point from where your directrix should be passing We should be the midpoint of n and s Isn't it? Isn't it obvious that this point has to be one two comma Yeah Clear everybody Okay, yes a line passing through one comma two Y axis has to be x equal to one my dear We're as good as saying x minus one is equal to zero Now what is left? We know the focus. We know the directrix. So if you take any point h comma k From s would be same as his distance from the directrix So h comma k distance from 5 comma 2 would be under root of 5 h minus And what is the distance of h comma k from x minus one? So it's simply h minus one mod by one. It's a simple formula distance of a point from a line formula Square with this h minus 5 the whole square plus k minus 2 the whole square is equal to h minus 1 the whole square Square it up h square h square go on. What will I have? I'll have k square I'll have minus 12 h I will have minus 4 k I will have 28 if I'm not wrong That's 28 One second sir. Yeah Anything which I'm missing please highlight Minus 4k plus 4 No, sir, no, correct Yes, sir. Yes, sir. So at square go on. I have minus. Oh, sorry minus 8 h will be there Minus 8 h will be there Minus 4k will be there and plus 28 would be there, correct me if I'm wrong No, sir Fair So if I generalize this my equation would look like y square minus 8 x minus 4 y plus 28 equal to Zero this would be the equation of my problem Now do you see any x y term? No, right? No, sir. It's not oblique. Yeah, you can see your axis is parallel to the x axis. It is there's no routine So whenever your axis is parallel to the the coordinate axis is On the coordinate axis is you will never find an x y term Is it clear? Yes, sir, but I'm not happy because I don't know if I'm able to solve it. So I'll be giving you one more. Yes, sir Sir, you use my dialogue The tension was that actually Okay Minus 6 comma minus 6 is at Minus 2 minus 2. Hope you have drawn the figure properly Yes, sir. Yes, sir. Wait, sir. Wait So it's an oblique parabola you'll have an x y term also appearing somewhere Sir, it'll be one second The denominator will be under root of 36 square plus 36 great Sorry, I didn't get that denominator Great That I'm getting h plus 6 whole square plus k plus 6 whole square is equal to h plus k by mod that by 36 plus 36, right? Did you get the equation of the directrix or when could y plus x equals zero y plus x equal to zero No, that is not correct Sir x plus y equal to four x plus y equal to four is correct So a parabola's equation is 2 x square plus 2 y square minus 2 x y plus 32 x plus 32 y plus 1 by 28 is equal to zero Yeah Most of it most of it is correct, but there's no 2 x square into y square. They're just x square Take it How? So this is your Axis equation, so this will be your somewhat like let me use some different color and Pink This will be your directrix isn't it the pink one is your directrix Yeah, I'm speaking Mac. Can't you hear me? Okay, now see it Uh, how did the equation of this directrix? Very simple the slope of the axis is negative. Yeah, so what is the slope of this line? Loop of the axis will be what? y2 minus y1 you can say four Four one by four, which is one So this guy will have a slope of Minus one or minus one correct Yeah At this point n would be that v is the midpoint of v is the midpoint of n and s isn't it So let n point be uh some alpha comma beta Okay So alpha minus six by two should be minus two So alpha should be Beta should also be equal to two if i'm not wrong Okay So this is comma two so let me write down the equation of a line Which is having slope one minus one and passing Yeah, can you hear me now? So x minus or my you can say y minus two Is equal to negative one x minus two so basically x plus y is equal to four Sir yeah, tell me uh my cut So it like on observation if uh the line The axis of the line passing through minus six minus six minus two minus two, right? That means y is equal to x in both the cases So it should pass through zero comma zero also, right? Okay, yeah So that means y equal to x line Is that thing and Actually passing through origin it doesn't look like in the graph. So it is actually the origin. Yeah passing through origin also, right? so uh And the other line also is perpendicular to that But it did not pass through origin, right? Oh, okay. Okay It can be anywhere parallel to it, but okay. Yeah, the distance is what matters. Okay. No Yes Yes, sir the equation of the direct rakes We got the equation of the direct rakes now. We are ready to use the locus definition so I can say uh under root of H distance from the focus K plus six the whole square Go to distance of h comma k from this line Which is h plus k minus four mod By under root of two getting the point Okay This side and square both sides. So you'll get two times H plus six the whole square K plus six the whole square Is equal to h plus k minus four the whole square Yes, sir. All right Hey, when you log in your computer, how do you uh, where all what all do you press? Call via Right something call via dial in via internet audio Once again dial in via internet audio, right? Yes, sir Dial in via internet audio Or device audio Okay, I know Okay, so now if we expand it by the way, you'll have two h square from here and h square from here That will leave you with an x square Again two k square and k square coming from here. So k square x y term will only be contributed by this fellow. So which is minus two hk So this will become a minus two hk Then you will have uh H term coming from here, which is if I'm not wrong 24 h From here minus 8 h so that will make you 32 h k term here will be 12 k And again from here you'll get plus 8 k. So which is going to be again 32 k if i'm not wrong And constant terms will be 36 or 36 72 into 2 is 144 but 40 from minus minus 16 that's 128 Your final equation after generalizing it would look like x square plus y square minus 2 x y plus 32 x Plus 32 y plus 128 equal to 0 Is that right? So how does the parabola change if x and y terms are equal and not equal? I'm sorry How does the parabola change if uh the coefficients of x and y terms respectively are equal and not equal Like in this example, it was equal previous example. It was not equal to 8 Huh It depends upon what is the line over here. So if this line has that's a 2k. It will not be equal Yeah, yeah So how does it look will it become fat then or it'll become oblique towards one um or something like that By the way, uh anybody who has max number Can you please give her a call and tell her how to join? Yes, sir. I'll do it early, uh, you know some girl adwetha or amla or chavi Could you please help her because she is not able to join someone She's saying she's not able to listen to me While joining that option comes on your computer, right? Yeah, dialing via internet over here One second, sir. I will do it Oh, thank you so much. Thank you Meanwhile, we can try one more question find the equation of the parabola with lattice rectum joining the points 3 comma 6 And 3 comma minus 2 Was it uh Was it is it done? Venkat is I'm dialing sir. I had to save her number from the group Okay, so I spoke to her. She's trying with the another device Okay, thank you so much Adweta Chavi sir Chavi yeah So you understand this question, uh Chavi yeah So you understand this question, uh the lattice rectum is joining these two points By the way 3 comma 6 will be here 3 comma minus 2 will be here I hope you all recall. What's the lattice rectum? Lattice rectum is basically a double goddess. God got that. Yeah, it should be perpendicular to the axis and passing through the These two points are given to you here So what's it how many parabola will be possible in this case? Uh one up down one up to down and one down to one will be like this from left to right and one other will be like this Yeah Two answers are possible Yes, sir. Oh, that's the question Okay Very important to know that in a parabola. What is the length of the lattice rectum? Okay for Like one second Like because it's vertical and stuff Okay, let me take a very very simple case where we have a standard parabola like this Okay, this is a lattice rectum passing through a comma zero Okay, can you tell me what is the end points of this lattice rectum? Of course since it is perpendicular Okay, yes, sir. Since it is perpendicular to the x-axis You can say that the x coordinate will not change for these end points, isn't it? I only need to figure out what are these question marks Now for that This is pretty simple All you need to do is in the equation of the parabola y square is equal to four ax Put the value of x as So it'll become four a square, isn't it? So y is plus minus two a correct Yes, I know So this will become And this will become Minus two a now can somebody tell me why I didn't take the upper one as minus two a and the lower one as two a It's always positive. It is a distance. Okay positive. That's right. Okay Oh in other words, what should be the distance between these two extreme ends of the lattice rectum? For a for a And so this is something which probably I did not tell you before Remember lattice rectum length is four a units or you can always say it's four times The distance between the vertex And the focus So one thing is very clear from this particular question that four a is equal to eight That means a is equal to It's a important information for us Why is it? It's up to you to figure out. Yeah, I can hear you now Mac Yeah, you need to mute mute your mic or let me mute you. Can you hear me now? Yes, sir. Yeah, thank god. Good Don't worry. The session is anyways getting recorded. So the part where you missed out you can always go and watch it Okay Sir, yes, sir. Tell me it's important because you can find out the coordinates of the focus Absolutely very very important. So let's say I take the case number one So let's say I have a parabola like this Okay, so now you know that This gap is going to be two Correct my dear Yes, sir And can I say that this point which is your Focus happens to be the midpoint of these two guys. So it has to be three comma Something comma two. Yeah, three comma two If I tell you to go two units this side, where will you land up? One three comma If you have to go left on this line by two units, where will you land up one comma one comma two, right So if you further go by two units, where will you land up minus one comma two minus one comma two Now directrix is a line which is going to pass through minus one comma two and perpendicular to this axis See at the end of the day, what do I want? I want focus Which I got I want directrix which I am going to find out now So now tell me what are the equation of the directrix? X equals minus one X equals minus one We all agree with him. I do Okay Now let's just do the no finishing task Now you know the process Just finish it off By the way, this thing I am going to erase it In case, okay, let it be like it So k square minus 8 h minus 4k plus 12 equals zero Sorry, sorry, sorry k square minus 8 h minus 4k plus 12 equals zero Okay, let me check now So Take any point Is distance on the focus should be same as the distance on the directrix So these two distances must be equal. Hope you can see the figure Okay H minus 3 the whole square k minus 2 the whole square Is equal to the distance of this point from x plus 1 equals zero which is h plus 1 mod Correct square both the sides So h minus 3 the whole square k minus 2 the whole square is h plus 1 the whole square Expand it One of the h squares will go off for a toss Correct So you end up getting k square That is this term. So I'm just cancelling out the terms which I'm taking into account minus 8 h Okay minus 4k Plus 12 equal to zero. Is that what you got Venkat Yes, sir generalizing it. It'll be y square minus 8x minus 4 y plus 12 equal to zero Very good But this is just one of the parabola Can you similarly find out the second case also that is when your parabola is of this nature Opening in this way that means let me change the pen color to Blue So when your parabola is of this way now your directix will be on this side Will the focus position change? No What will be the directix equation? One second Go to on this side and again to on this side Seven comma two Seven will be your line. Yes Sir, can we just reflect it about this line? Which line media? One second sir, three comma six three comma minus two. It's just getting reflected there. Yeah, we can do that Yeah Oh Sir y square minus 4 y minus 36 equal zero That seems to be correct. Let's check So your answer would be again i'm cutting Down the steps So x minus three whole square Is equal to h minus seven the whole square. Am i right? Yes, sir Expand both the sides x square x square gone So we'll have k square. Let me write it y square directly Uh, you'll have a plus eight x and you'll have minus four y And 13 minus this is minus 36 equal to zero. So this is going to be a second parabola. So these two are your answer Everybody how it works? Yes, sir Okay, now we are going to move towards the three other standard cases. So now you looked at a parabola, which was Your parabola opening to the Left sorry right, isn't it sides? Okay. Yeah, which was your this parabola Isn't it? So this was the case where your parabola was opening to the right Okay, and this equation was y square is equal to four ax, isn't it? by the way Vertex was at origin focus was a comma zero Directrix was x equal to minus a Later sector length was four a Equation of the axis was y equal to zero Okay tangent at the vertex was x equal to zero etc. So everything you should be knowing about it guys Whenever y square is equal to four ax comes in your mind this image should directly Register in your brain. Okay. Now that this is just one type of standard case There are three others. So let's look at the second one in second one your parabola is going to open to the left Something like this Now here the vertex is taken as minus a comma zero Okay Directrix becomes x equal to a line Vertex, however still remains at the origin Okay Length of the latter spectrum is still four x Later spectrum length doesn't change Because you're not feeling the dimension. You're just flipping about the y axis Okay Axis equation still remains y equal to zero And tangent at the vertex also remains x equal to zero Now, can you can you tell me what would be the equation of this? Y square equals minus four x minus four ax y square is equal to minus four ax See simple when you're reflecting a graph about the y axis you flip the sign of x That's it or you put x as minus x Third case is when your parabola is opening upwards Like this The one which you have been using in the quadratic equation also. Okay This parabola will have the vertex at origin Okay, focus will be at zero comma a Equation of the directrix will be y is equal to Minus a length of the latter spectrum is still four a units. Can you tell me the equation of this? x square is equal to four a y Absolutely x square is equal to four a y now. How did you reach to this conclusion? That is x square is equal to four a y So replace x with y Yes, basically you would see that This third case could be obtained from the first case by reflecting the graph about y No, no, no about y equal to x line. Yeah, that's right So I'm just showing a miniature version here see If this equation this graph If you reflect about This line y equal to x line see what will happen This part you see the protruding part of yellow. It will become like this. Okay This part you think this part will become like this This part will become like this That means you'll end up getting this graph Okay, so again recall your age record your uh bridge course days Where if you are swapping the position of x and y you are reflecting the graph about y equal to x line Plain and simple Yes, sir Old case When your parabola is looking downwards Sad Yeah, it's that face in this case. This will become zero comma minus a as your focus direct vertex will remain zero zero Direct x equation will become y is equal to a And later sector will be still four a units and this time the equation will be x square is equal to Minus minus four a y Okay, just a quick question for that so that everybody's on the same page find number one vertex Focus equation of the direct x equation of the axis length of the latter spectrum for these two parabola number one Y square is equal to minus sixteen x and number two x square is equal to minus y Quick you should not even take one minute to solve this Vertex zero comma zero Focus is a red so Wait, sir. Do you want how do you want me us to tell sir? coordinates Like one okay, sir So for the first question zero comma zero minus four comma zero x equal to four x axis and sixteen Everybody's got the same So the moment you see this what image will come in your mind I will always imagine a parabola opening like left words like this. Happy pungal to everyone Okay, focus will be Minus four comma zero has everybody got this site Yes equation of the directrix would be x is equal to a so x is equal to four got this Equation of the axis will be x is equal to sorry y is equal to zero x axis Yeah x axis x axis equation is y equal to zero. Okay length of the latter spectrum is this number 16 units Everything The moment you talk about the second guy this image will come in your mind. Yeah, like my face. Yeah So I light it in site is far apart So the second one vertex will be still zero zero zero comma zero Then zero comma minus one by four. What is he? Hey, hey, what are you? Hey Minus one minus a one by four Never say minus for a Venkat, please. Yes, I realized a has is always positive is always positive remember that Yes, sir Because a signifies the distance distance of the focus on the focus Yeah, so now tell me zero comma minus a right Yeah, this is your focus equation of the directrix y equals quarter y equals quarter, okay y axis because the axis is a y axis, which means x equal to zero L. R. Is my j rank Thank you, sir Okay, this was the standard case pretty easy standard is what is going to be asked in your school, uh, you know exams But unfortunately life is not as easy. So we move on to something called as the shifted parabola shifted What are the plural of parabola? Parapole Parapole and parabolas both are fine. We also call this as the generalized form My spelling of generalized is always wrong generalized form shifted means In all the previous standard forms that you saw your vertex was always at origin, right? Whether it is right opening left opening up opening down Vertex remain at origin, isn't it? Now in case of shifted versions your vertex will no longer remain at origin It will be at some point at the alpha comma beta But we'll ensure that the axis of the parabola still remains parallel either to your x-axis or your y-axis Right, the shifted cases are those cases where vertex Is not at the origin But the axis of the parabola still remains parallel to either the x-axis Or the y-axis depending upon the scenario So now I have a question for you all Let's say this parabola y square is equal to 4ax which I'm showing with a dotted yellow line Goes to this position without any change in the structure That means there's no you know change in the dimensions Now the vertex here comes here By the way the axis is still parallel to your blue axis At this point here is alpha comma beta Can you tell me what would be the equation for this white parabola? Given that yellow parabola is y square is equal to 4ax So much bridge course you have done shifting shifting and all Hmm One second y square equals 4ax so it can beta units up So y minus beta whole square equals 4ax minus alpha So Venkat is saying y minus beta whole square Because 4ax minus alpha is he correct everybody please verify this Is Mr. Sai Venkat is correct Say yes print a type yes if he is correct or type no if he's not correct Do something you guys you're you're not doing any physical work today Physical work So did you know Sai in Canada means die? Yeah, yeah Oh Sai means Sai Baba Sai means die also. Yeah, we are all dying every day Right. Yes. No, he's correct. He's wrong. Yes, sir Okay, when I agree with him, he's right. Okay, by the way When you are getting this equation Where are you shifting your origin to? Minus alpha minus beta absolutely correct. That's something which you know you got it right this time When your parabola has gone up and right remember your origin was actually shifted to Left and down which is actually minus alpha minus beta That's why your x gets replaced with remember whenever you're just trying to recall a bit of a theory I'm writing it over here When you yeah h comma k remember your small x should be replaced with capital x per se So here your x would be replaced with x minus alpha y will be replaced Because it is minus alpha and why So this x that you see over here just got replaced with x minus alpha and this y got replaced with y minus beta But normally we don't write, you know our equations in capital alphabet So we made it small and this is what Sai told and he's after Okay Yes, sir, I have a question for you Yes, sir Doesn't I use redboard sir Redboard, huh? Come on some fun in the class redboard. Yes, sir Everybody likes it Redboard with pink So y minus four whole square Is equal to eight x plus three. Okay, so let's say this is a parabola equation. Okay Find number one vertex number two four number three equation of the direct x Number four equation of the axis Number five length of the lattice rectum You may take the help of a sketch to do that I'll give you exactly two minutes for this Let's discuss after two minutes Vertex will be four comma minus three it should be other way around my dear minus three comma four right minus three Sir Answer sir Vertex minus three comma four that guy's Now if you draw this sketch It's basically a case where your parabola has been shifted And the vertex is now gone to minus three comma four minus three comma four means you are here correct so now minus three comma four And opening right towards something like this. So vertex is minus three comma four undoubtedly Now where is your focus? zero comma four. Now remember This information hides the information of four a That is a is equal to two two. You have to go two to the right from there Yeah, yeah So, yeah, three comma four. So now you will reach minus one comma four, isn't it? Yes, sir. The focus is that minus one comma four Correct Directrix will be aligned parallel to the y-axis and passing through go two units to the left from here Where will you? Yeah Minus minus five comma four. Yeah, can I say equation of the directrix will be x is equal to minus five Or you can say x plus five is equal to zero both mean the same thing Yes, sir Equation of the axis will be y is equal to four four and eight And length of the latter sector will be eight units. How many of you got everything correct? I do one silly mistake equation directly into x equals minus five y equals minus Now many people say sir for solving such questions. Do we always have to sketch it and figure it out? Answer is no. You can do it without sketching also and for that I'll tell you a trick The same question I'll solve it without you know imagining this graph Okay, sir. How let's see y minus four the whole square is equal to eight x plus three I will compare it with a standard case and the standard case has to be y square is equal to four ax That's all Here there's a rule change over here. Your capital y is playing the role of y minus four Okay, so we can say this and this are same terms. Okay, your capital x is playing the role of x plus three x plus three And a is basically two over here two No, I've hidden the diagram so that you can't see that your mind Just imagine this Close your eyes and imagine this parabola Okay, hope you're imagining it Yes, sir. That is the vertex zero zero, isn't it? So make something like this x equal to zero y equal to zero instead of adding zero zero directly just do some x equal to zero Now replace capital x with small x plus three replace capital y with small y minus four So end up getting x as minus three y as four So your vertex will become minus three comma four Is it what we got here? Let's check. Oh, yes, we got the same thing. Again. Let me hide it Next so vertex hole is over Now for focus again close your eyes. Where is the focus of this guy? A comma zero, right? Yeah, so write x as a and y as zero a comma zero Don't write just write x as a and y is zero Okay, again do a roll change capital x is x small x question is how much to So from here x will become minus one y will become four So minus one comma four will be your focus. Let's check whether we got the same answer. Oh, yes, we got the same answer Okay There's no need to sketch any graph next equation of the directrix So directrix equation imagine in your mind it was x is equal to minus a The less we're all change equal to one x plus three is equal to minus two one Oh, yeah, yeah, sorry equal to zero check. Is this the answer? Yes, sir. Yeah x was five equal to zero. You can see that next Access equation for this parabola was y is equal to zero correct So your answer would be y minus four equal to zero. So this becomes the equation Yes y minus four equal to zero And yeah, let us take them, you know, four a That's just good thing Because yes, sir All the type of shifted case without sketch Yes, sir Now try another question, which I'm giving you Give me all the five things Can I go on to the next page? Yes, sir Which color page you want blue blue? Yeah, so now my parabola would be let's say X Minus three the whole Is let's say minus 16 y plus Two Again same thing I want. What are the same things vertex? Equation of directrix equation of axis Length of the lattice victim Don't sketch the diagram Yes, sir, sir Sir, can I get the answer? Can you tell you privately? Text you like here A private text. That's me privately Okay, sir. You're sentiment academy Hey, come on I'm texting in order only, sir. Oh, okay. Directrix one second, sir. Oh one second one second, sir Yeah, so Manav All of you I'm seeing your answer. Don't worry. Keep typing Are we ready to discuss it now? Sir ignore my third message, sir This is And then we'll see and then we'll see who all have got the right answer. Let's check it out So the first thing is The moment I see this parabola what image I will bring in my mind. I'll make the image of x equal to minus four a y my dear friend Okay, because of this minus sign I have to assume it because a cannot be negative Yeah I can't assume x square is equal to four a y because then I will be claiming that a is negative four Which is wrong. A can never be negative Yes, sir. Who's playing the role of what? So capital x whole is being played by x minus three capital y roll is being 1 plus 2 a roll is being played by four Okay, in your mind imagine this parabola Imagine this Yeah So here you know that the vertex would be at zero zero So put x as zero y as zero that means with small x minus three minus three is zero And small y plus two is also zero x is three y is minus two so three comma minus two is your vertex three comma minus minus two Yes, sir So this was your vertex for focus You know that for this parabola your focus is at zero comma minus a So right x is zero y as minus a yes or no So zero will be equated to small x minus three and small y plus two would be equated to minus a which is minus So x becomes three y becomes minus six Right, so three comma minus six is your focus for direct tricks For direct tricks again, imagine This parabola where is the direct tricks y is equal to a y plus two will be equal to will be equal to four So y is equal to two is your direct tricks getting the point Yeah equation of the axis For the equation of the axis, we know that it is going to be your Y axis, isn't it? Yeah, this means x is zero That means x minus three equal to zero is your equation of the axis Length of the latter sector is four a four a is nothing but 16 units. How many of you got everything correct? None of you No Everything correct You one more or are you good with this? One more but one second Yes, sure So for this test still where in parabola is it coming? Uh, not very heavy concepts very light concepts related to equation forming of equation and not Okay, I've purposely kept it light because I was not sure till what level I'll be able to complete it But yeah, you can manage it with this information Oh, yeah, peace which color? Uh, great Okay, you send the combination yellow and black Okay, so since you were asking about one more question, let's take another question Let's take Y minus four whole square is equal to negative X plus one Again the same old things we want here the vertex the focus the directrix Equation the axis equation l r Mm-hmm Oh, sorry, sir. That's wrong. Yes, sir Others ready to go? Uh, one second one second. Yes, sir. Should we discuss? Yeah Okay, time up. Let's discuss now Which parabola will you compare this with? Y is equal to minus four a X left side open Okay. Yeah, let's do a role change. These are the roles played A is one fourth. Okay. So the moment you do this imagine The graph in your mind. It's the one way which is left side open like this. Okay now for vertex for vertex That zero zero is the vertex. So X is zero Y is zero That means X plus one is zero Y minus four is equal to zero That means minus one comma four Minus one comma four is your vertex Hope you have got this correct focus would be Zero comma sorry minus a comma zero. So X is minus a Y is zero So X plus one Minus one fourth And Y minus four is equal to zero. So X will become minus five by four Y will become four So focus is minus five by four gamma. Okay Directrix equation is X is equal to a if I'm not wrong Correct. So X plus one is equal to one fourth. That means Directrix equation is four X plus three equal to zero X is axis equation axis for this is still your Y is equal to zero. That means Y minus four equal to zero The length of the lattice sector Sir sir So here it was X equals zero Y equals minus a if it was X square equals minus four A Y So you're shifting these guys accordingly, right? Directrix A comma zero zero comma and all That's why I asked you to imagine them, right? The moment you imagine them, you'll you'll start writing these basic ones Yes, sir Don't expect the world to treat you in the same way as I am treating you They will not give you something like this since I'm a kind hearted person. I have given you like this But you are the world sir, actually How would the day people give you this question? They'll give you like this They will say They'll the parabola whose equation is X square plus 8 X plus 12 Y plus four equal to zero find number one vertex Number two focus Number three equation of directrix number four equation of axis Number five length of lattice sector This should be X square something something X square equals The first thing is what are you going to do here? Since there's no XY term Complete the square. Yes complete the square wherever it is necessary Quite a bit will help you after this After you try it Are you okay? Wait, sir Sir, I sent you one thing Like after this is the same cute process. So it's fine. Okay. Let's check So here Since you see an X square term It's very obvious that there would be a perfect square getting formed in X. Correct So what I'll do is I'll first write it in this way Okay, I'll keep these separate And this is nothing but coming from X plus four the whole square minus 16. Isn't it? Yeah Which is nothing but X for the whole square Is equal to take everything to the side that will become 12 minus 12 Y But remember our effort should always be writing it as something like this X plus minus alpha whole square is equal to plus minus Y plus minus beta, isn't it? So what is in order to make the coefficient of Y indirectly one I will take a minus Something like this Okay, this was the form which I used to give you the questions But now you have to bring from this form to this Now once you know this form you know that it resembles X square is equal to minus four A Y Yes or no Now who's playing the role of what X plus four is playing the role of capital X Y minus one is playing the role of capital Y And three is playing the role of A Now imagine this parabola. Sorry this parabola in your mind And start writing stuff that you know for vertex, you know, it's zero zero. So no doubt about that So my vertex will be Zero zero means X is zero Y is the minus two Four is zero Y minus one is zero So minus four comma one is your vertex Okay, yeah For this parabola zero comma minus eight So write it as X zero Y is equal to minus A So X plus four is zero Y minus one is equal to minus of three So minus four comma minus two is your focus directrix Directrix, you know for this parabola is Y is equal to A that means Y minus one is equal to three That means you can say Y minus over is equal to zero Okay Axis of this parabola is your X equal to zero. That's your Y So basically X plus equal to zero is your axis And later take the length is 12 minutes Let's check the answers that you have given Okay Ambulance some mistakes have happened this check Is that for this question only a previous question? This question only Let's check it out Okay, guys Yes, I've done for sure Yes, sir The next concept that we are going to talk about is the concept of position of a point Oh, sorry about that parametric equation of a parabola parametric form or parametric equation Start with Y square is equal to four AX. This is a Cartesian equation as you all know Okay So we we can write a parametric form for this as you all know parametric form is basically a form where you write X And Y in terms of a parameter But if you eliminate the parameter, you'll end up getting the same equation back Okay, so for this parabola the parametric form is given as X is equal to AT square and Y is equal to 280 where T here is a parameter Okay, so try eliminating your parameter T. You would realize you'll end up getting Y square is equal to 4x as your answer Okay, what is the use of a parametric form? It facilitates us to choose a point on the curve So basically if I have to choose a point on this parabola Y square is equal to 4x I will choose the point as let's say AT1 square comma 282 AT1 Okay, I can choose another point. Let's say AT2 square comma 282 Okay, I can choose another point AT3 square comma 283 so every point has a different parameter for it Okay, so this is this parametric equation helps us to choose points on the curve Now question for all of you Can you suggest me a parametric form for this parabola? I think in similar lines Sir Y equals minus 280 One second Give me that sad parabola One second sir, it's a leftward opening You can write it You don't want to speak out. You can write it down privately to me X equal to minus AT square Brilliant X equal to minus AT square Y is equal to 280 Okay Tell me for X square is equal to 4AY quickly Sir X equal to 280 Y equal to AT square Absolutely correct Michael, very good Next tell me for X square is equal to minus 4AY Replace 280 minus AT square X is equal to 280 minus AT is very good Now take a clue from this and suggest me a parametric form for X plus 3 the whole square is equal to minus 4 Y minus 2 Sir could you just go back one second? Yeah, sure It's screenshot Can I go back to the question now? Sure, sure, sure Yeah X equal to 280 minus 3 Okay, by the way here A is 1 you can use that 2D minus 3 Okay Everybody is done by the way Give me X and Y both Are we ready guys? Yes sir Okay See guys don't get scared you just write for whatever you have learned to now which is let's say if your parabola is like this What is the parametric equation? X is equal to 280 and minus 3 and Y is equal to Minus minus AT square So we start replacing things The capital X is X plus 3 280 means 2T Capital Y is Y minus 2 minus AT square is like minus T square I can write this as X is equal to minus 3 plus 2T and Y is equal to 2 minus T square So this is the parametric equation Okay, however, this may not be required for you but just for your information Okay Sir how is this useful See if I would choose a point on this parabola I would choose it like this I would choose a point like minus 3 plus 2T Minus 3 plus 2T comma 2 minus T square See I don't have to choose two separate variables Oh, okay, okay Working with lesser variable is always convenient Sir, but how did this parameter be born sir? See again, it's not like, you know, there's a fixed parametric form You can suggest your own parametric form also Remember in circle I told you there are various forms for the same circle Yeah It's just like thinking backwards. So how could you choose a point which suitably fits that equation? Oh, sir could you give another example of parameter for parabola like something? See let's say if you say X Y square is equal to X parabola, okay Doesn't this remind you that I could choose a point as T comma T square? Yeah Very natural, but somebody can choose something like this T by 2 comma T by T square Somebody can choose T square T to the power 4 also Yeah What all of them are parametric form for the same parabola, but what is most convenient to use? This is most convenient to use. Okay, so you can say Yes, sir, that's what I'm trying to say. Yes, sir Yeah Next we'll talk about position of a point After this, I'll take a small break for 10 minutes. You can go and eat something position of a point With respects to a parabola. So let's recall our friend Now a point X1 Y1 by the way, this is our standard form Now let's say there's a point P X1 Y1 This point we can lie in three positions. Either it can lie within the parabola like this Or on the parabola like this Or outside the parabola like this Okay, how would one figure out whether it is at position a or position b or position c? That depends on the half of lattice victim length See, I think you you think that it is located at on the lattice victim or nothing like that It is okay What is the time? Slightly deceiving, but it's not located on the lattice Oh, one second. I will think So if it's b that it should satisfy that equation, whichever it has Okay, very good So it's very obvious that if it is at b then y1 square minus 4ax1 would be Zero equal to zero Then a should be less than zero Very good. Why why should be less than zero? Wait, sir I solved by symmetry It's very simple. See all of you please focus here Can I say the y coordinate at b Would always be higher than the y coordinate at a Yeah, here this guy's dominating there the other guy will dominate Absolutely. So here y square is exactly balancing out 4ax1 For the moment a y square becomes smaller in magnitude while x more or less doesn't change Isn't it? Yeah, that means this guy will not be you know I would say less than zero Suppressed by this guy. So y1 square minus 4ax1 will be less than zero And it's very obvious that C will be y1 square minus 4ax1 is greater than zero. Yes or no? Yes, sir Yeah, do you support cab? I don't know Be neutral Okay So this is something where I'm trying to bring in the concept of a circle Remember in circle, we used to have something like s s1 t and all Yes, sir. The same thing will be there in parabola as well and ellipse also hyperbola also In this case, you know, what's your s y square minus 4ax that is called s Oh, that's why y1 square equals minus 4ax1 equal to zero is the equation there also s1 equals zero Yes, there s1 will be this expression. Remember s1 is not any equation s1 is a number What number the number that you obtain when you put x1 y1 in s Okay, so if that number is equal to zero that means the point which you substituted was lying on the parabola If the number that you got was negative that means the Substituted was within the parabola. So you can say you can call this as s1 equal to zero You can call this as s1 less than zero and you can call this as s1 greater than Okay Now, let me tell you s1 s etc They will all depend on the curve itself. So for this curve, it is your s But let's say if I give you a parabola like this x square plus y square minus 4x plus 2y plus 3 This will become your s my dear Getting my point Yes, sir Parabola to parabola Okay, now I have a question for all of you Yeah, find the position of Find the position of minus 2 comma 2 With respect to the parabola y square minus 4y plus 9x plus 13 equal to zero Then C Is it C? Are you mean lying outside the parabola? Yes, sir. Okay, let's check So this is our s term I'm substituting the point here. So 2 square plus 8 Uh minus 18 plus 13 Guys, this is clearly greater than zero. So if it's greater than zero It's going to be How much is it? I mean greater than zero on here, right? Yeah, greater than zero. Yeah, greater than zero. Yeah Yeah, greater than zero. That means it is outside the parabola. Okay Now let's have a break. Yeah, sir Uh, like I've observed one thing I did this using completing the square We I did this using completing the square Uh, and we did another problem using completing the square So I just saw there's some magical force every time you complete once again I'll use my tensor No, no, no, sir. Wait that page only So every I observed some magical force being here. Uh, every time we complete the square here Uh, this term and this term the coefficients become the same like how is it? No, you get it as Oh, okay. Okay. In this case, well it is happening Oh, okay. A few pages coincidentally it is happening Oh, okay, sir break Can we resume at 11 25 am Yes, sir So next concept that we are going to talk about is something related to We talked about parametric equation right now. So we'll talk about The parameters at the end of the focal chord. So let me give you as a question So let us say this is our standard case of a parabola This is my focus And what I do is I make a focal chord Okay Now the focal chord is basically connecting two such points Whose parameter is that say t1 and t2 respectively now when I say t1 t2 respectively what I mean It's just a way of saying that this point is a t1 square comma to a t1 and this point is a t2 square comma to a t2 Okay now prove that t1 t2 is minus 1 that t1 t2 is minus 1 sir. Are you free till 130? I have a class at Okay, are you free till 125? Yeah, yeah Okay, sir Sir, I have an idea, but I'm not sure if it's acceptable or not Let the two Points be the opposite ends of ladder Okay, this is not a ladder symptom. Okay. It can be a ladder symptom in case your focal chord is perpendicular But why were you Using oh, okay. You need to prove it, right? Yeah Sir, I still didn't understand what exactly t1 and t2 are. I know that points and stuff but Yes, what is the what is the idea? Sir, I didn't understand exactly what t1 and t2 are t1 and t2 are parameters of these points Yeah, every point has a different different parameter. Yeah, yeah that I understood. Yes You have to connect the parameters of these two points and prove that they are related to each other by Okay, okay. It's a one second. I have an idea See Venkat, it's just sufficient for you to understand that if I say t1 is a parameter for a point Then this point is actually a t1 square comma 281 And let's say t2 is a parameter of a point then the point is a t2 square comma 282 So we normally write it in a shortcut way like this Oh, okay, sir. Oh, okay. Yeah t1 means a t1 square comma 281 t1 is a parameter of this point which means the point itself is this Oh, okay. Like we say eccentric angle is theta in a circle and that point is a cos theta comma a sin theta, right? Yeah, the same way here Okay, sir One second, sir. I have something. Let's see Don't like to go too. Just focus on these coordinates Yeah, that's what I'm trying to uh Like connect a line and stuff Are you able to prove? Wait, sir. One second x minus Oh, sorry, uh, Sita is not there So one second a minus a t1 square equals t1 by t2 into minus 281 one second, sir a into one minus t1 square equals a into t1 minus t1 square by t2 t1 square minus 1 One second, sir. I almost got a t1 square t2 minus t2 I mean, is this problem so difficult? No, sir. I'm almost done. So I just got a 2 t1 square equals t1 square t2 minus t2 After that, I don't know how to proceed Let's take all this point as s. Do you realize that The line passing through those point passes through x-axis intersecting at a comma zero The slope of a s would be the slope of b s Yeah, what is the slope of a s? 282 by a t1 square minus Similarly, this will be a 2 a t2 a t2 square minus 1 Correct. Yeah a a 2 a 2 a go on cross multiply So t1 t2 square minus 1 is equal to t2 t1 square minus 1 So t1 t2 square minus t1 square t2 is equal to even minus t2 Take t1 t2 common you'll get t2 minus t1 Here also you can write it as negative of t2 minus t1 Now since you know t1 and t2 are not the same points t1 t t1 minus t2 is not zero so you can actually talk And then you'd end up getting t1 t2 is minus 1. This is property. So please remember it The parameters at the end of the focal chord are basically multiplied to give you minus 1 That means if I know this point a t1 square component to a t1 The point will automatically become Then the other point will automatically become a by t1 square comma minus 2 Are you getting my point? Yes, sir. So I have a question very small question sir. I have a doubt. Yeah Sir, uh, you used slope of a s equals slope of b s right? Yes Sir, can you also do by uh saying the line passing through these points? Intersects the x-axis at a comma zero Okay. Yeah, we can do that. Why not? Yes, sir. I was doing that and I got stuck. Could you help me out? See here First of all you have to find the equation of a line passing through a b, right? Yeah. Yeah that I did So let me write it like this. It's a longer way, but say that I'll do it So y minus 2 a t1 Equals t1 by t2 Why? Del by by del x Del y By del x Yeah, by del x x minus 2 x minus a t1 square, correct? Yeah, yeah, yeah, which we're actually writing as y minus 2 a t1 is 1 by sorry 2 by t1 per c2 Yeah Why are you saying a comma zero should satisfy it Zero minus 2 a t1 Yes, sir. So minus t1 times t1 plus t2 is equal to 1 minus t1 square So that's minus t1 square minus t1 t2 is equal to 1 minus t1 square minus t1 square minus t1 square gone So t1 t2 is minus 1 Okay, sir Okay Yeah Okay Now remember when I was talking about ladder system. I told you ladder system happens to be the shortest focal chord Yes, sir. Okay. Now the time has come that we prove it With the shortest so now let's take Our age old friend Y square is equal to 4x now Let me draw a focal chord Now I would like you to tell me what is the length of this focal chord. Uh, let me assume that this point here is A square comma 2 a t1 And this point here automatically will become A by t1 square a comma minus 2 a t1 Right by t1. Correct. Now in terms of a and t1 tell me what is the length of the focal chord ab Apply distance formula Whether distance formula or any other method tell me what is the length of ab Okay, sir But give it a simplified way. Don't just write a crude expression and Yes, sir Sir under root a t1 4 minus a Sorry, sorry a under root a t1 4 Plus 2 a t1 cube plus 2 a t1 minus a by t square Or what if I tell you Venkat that the answer is simply a times one t1 one by t1 whole square Would do You know you you actually found out the distance formula, right? Yeah, but you forgot the fact that Okay, so can I say this distance should be equal to this distance? Yes or no Okay, and similarly bs will be equal to bm That means this distance is equal to this distance and this distance is equal to this distance So ab is actually m m down as is the ab is actually as plus sb Correct and as is am and sb is bm correct Now am is basically This distance Plus this distance right, you know, this is a and this is a t1 square So am is a plus a t1 square Similarly, bm will be this distance plus this distance Take a common so which actually becomes a t1 plus one by t1 the whole square Yes or no Okay Now if you go back to the chapter series sequence, we have learned that am Is always greater than equal to gm, right? Okay, so can I say t1 plus one by t1 by two will always be greater than t1 into one by t1 under root So t1 plus one by t1 will always be greater than two Correct. That means this term will always be greater than a into two square, which is four Which is nothing, but it will always be greater than four a which happens later Okay, so any chord will always be greater than equal to the length of the latter sector Which indirectly means what that the latter sector is the shortest vocal chord So it also can we use the shortest distance from any point of a line is the perpendicular distance lattice distance is perpendicular No, no, no Why not because You're not finding the shortest distance from anything. You are basically playing with the length of this chord Yes, sir. You're increasing this side. This side also is getting reduced, isn't it? So, where do you strike up? Oh, yeah, where do you strike a balance? So that's where it is perpendicular. Oh, yeah Now I have another question for you all related to this only if this length is l1 And this length is l2 Okay Prove that one by l1 plus one by l2 is one by Yeah, wait one second not sure but I got I might So So two minutes more set this yeah, yeah, then anybody else who who's who's done with this Sir, I got it very good You don't take much time because I've already done half the work for you I've already discussed with you that l1 length was actually a plus a t1 square Right and l2 was a plus a by t1 square, correct Isn't it what we had discussed l2 is this length L l1 is am length correct Now if you just write this term as a plus a t1 square by t1 square, okay And just write one by l1 and one by l2 addition So you'll end up writing This So one plus t1 square you can take common This and this gets cancelled is equal to one by correct Now there's something very interesting over here If you write this as One by l1 plus one by l2 as one by a you would realize a is l1 l2 By l1 plus l2 isn't it? Then yeah over here Does this remind you of something in your sequence series top harmonic progression, right? So we say that 2a l1 and l2 are in harmonic regression So you can say a semi lattice rectum Why semi because today Is 4a is lattice rectum Yeah, is the harmonic mean of l1 and l2 this has directly come as questions in your cognitive experience Sir I had another idea Yeah, tell me Like half the time my ideas never work. But anyways, it's fun as I can you go back to that figure Okay, could you push the board up a bit? Here? Ah, yeah, yeah, so sir What I did This conveying it to you Yeah So I produced that there. I'll be defacing the figure for some time Produce a figure there. Okay And try to apply similarity of triangles Will it work? It should be but uh, yeah, it should work then. What do you do? Oh, yes, sir. I started comparing uh, once I can change colors This side This side By This side Okay, uh, is equal to this side by that side Equal to this side by this side It's doing that. Uh, when we started discussing it. Yeah, it should work. It should work right out Right now No, not now right out when the class is over Okay, sir. Okay. Okay. I have a question for you all Show that The focal chord of a parabola Y square is equal to 4x Which makes an angle alpha with the x-axis Which makes an angle Alpha with the x-axis is of length Is of length 4a cos x square alpha So hope the question is clear So what it tries to say is that If you have a focal chord Okay passing through the focus Making an angle alpha with the x-axis then it's length That is your ab length would be 4a cos x square alpha Can you prove it sir? I'm getting us 4a cos x alpha No cos x square Not cos x square Anybody else who's facing a similar problem? No sir, I'm wrong sir Should I help you guys? No sir, wait Sir, is the lattice rectum in some way related to the biological rectum? No No, okay Okay, all of you see here Let's say I call this point as a t1 square comma 281 Okay, now recently we have learned that the length of the chord ab will be what a t1 plus 1 by t1 the whole square, isn't it? Yeah Okay Now tan of alpha tan of alpha is the slope of Yes, isn't it? y by x Yeah So can I say this 2a t1 By a t1 square minus 1 Which is nothing but 2t1 by t1 square minus 1 Which is nothing but 2 by t1 minus 1 by t1 Okay Can I say then t1 minus 1 by t1 is actually 2 cot alpha? Correct Yes, sir Now this guy ab This guy ab Can I write it as a t1 minus 1 by t1 the whole square plus 4 This t1 minus 1 by t1 I'm representing with with 2 cot alpha so 4 cot square alpha Because 4 outs it will become 4a cot square alpha plus 1 I want cot square alpha plus 1 It's very easy when you do it sir Which one sir can do 81 I suppose if you remember you'll save a lot of time this length of a focal chord Passing 7801 square comma 281 Is that fine can we move on? Yes once a cot square alpha plus 1 yes it done Next is intersection of a line with a intersection of a line y equal to mx plus c With the parabola y square is equal to 4ax again. Let me draw the old parabola with us Now a line let's say y equal to mx plus c Hello, sir. I'm not able to see the screen Not able to see the screen. What about others? No problem sir There's no problem with others. So you need to join again Yeah, sir. Okay fine. It's your personal internet problem. Yeah Let's say y equal to mx plus c can have You know can interact with this parabola in three ways either it can be a secant line Or it could be a tangent line And neither secant nor tangent I don't know So here case number one. It's a secant line case number two. It's a tangent line and case number three It's a ns nt case ns nt means neither secant nor tangent Okay So ns nt case. This is a t case and this is a secant Okay Now I would like you to tell me For each of these conditions one Two and three What is the relationship between mc and a? So what should be the condition holding true between mc and a such that the given line y equal to x plus c is a secant to this curve Or a tangent to this curve or neither secant nor tangent to this curve Hey, sir I'm not sure sir, but this is my answer. Uh, if m square is equal to four a Okay Then it is a tangent Sorry m square is equal to four a Four a yeah Is a tangent. Why is it correct, sir? No But I want to know the reason why Okay, sir. So what I did I just first step I chuck this constant out this one See guy I removed the constant then I squared on both sides y square equals m square x square Oh, okay, y square equals m square x square Then I compared with this equation Then equated it then it should fall on the line because it touches it. That's a Can you repeat once again? I didn't know it's quite insane that Yes, sir. So, uh, yeah, so I removed this constant first of all see After that squared on both sides And compared the equations Why you removed the constant from from the side? So you wanted a relation between Uh MCNA Oh, okay, like yeah, okay Okay, if I were you see this is a case which is slightly different from a circle in circle. There was geometry to help you correct That means you want condition number one That means you would say distance of the center from the line should be less than the radius, correct If you want condition two to hold true, then you'll say distance of the center from the line to the radius And if you want condition number three, you'll say distance of these The line should be greater than the radius But there's no radius in case of a parabola. However, there is something called radius of curvature and all which is not related So how do you find in the case of a parabola the condition? For that your quadratic equation will come to your rescue What I'll do first I will try to simultaneously solve this line and this parabola by eliminating your y If I can say mx plus c I'm replacing my y with mx plus c This will be m square x square And this will become two mc minus 4ax Plus c square equal to zero, isn't it? Now can I say first question number one to happen this quadratic must give me real and distinct roots Oh my god, yeah That means your discriminant must be greater than zero Yes, sir c minus 4a root square should be greater than four ac Yeah After of two from both the sides, sorry four from both the sides, this should be greater than m square c square Square it up Listen, this will go off So 4a square is greater than 4amc That means a should be greater than mc. That means c should be less than a by m Please remember this So for situation number one to happen c Less than a by m first question number two to happen c must be equal to a by m and situation number three to happen c must be Greater than a by m that means it is equivalent to saying That discriminant is greater than zero here. You're saying discriminant is equal to zero and here you're saying discriminant is Not existing Discriminant is less than zero means roots are not existing, correct? Yes, sir. One second, sir So can you go down for a minute? Yeah, yeah this point Yes, sir Sir, do I spoil all your all sentence recording by constantly speaking? No, not at all. Your question helps many other students Oh, thank you, sir. But anyway, no one asks, it's rectum related One, two, ten, sir. Yes, sir. This condition has to be Remember this is called the condition of pregnancy Do you remember we had a similar thing for a circle? Yes, sir. What was that? That equal to zero What was the condition of pregnancy for a circle? Wait, sir, I am looking. Ah, if you're looking means you have not done problems Yes, sir. I am not done It was c squared equal to a squared into one percent Oh, yeah, yes, sir Okay Very important statement from me here is This condition is disclaimer Okay, disclaimer Only was standard from something Yes, these conditions are only Not even all the standard forms. Only this guy it works. Y squared is equal to 4x Oh, okay So I have a question for you with your Y2 to mx plus c Is a tangent to Y squared is equal to 4x Then your c is equal to a by m, right? Yes, sir. What will happen? To this condition Is a tangent to y square is equal to minus 4a. What will be the condition? I'll give you one more constraint. You can't use your quadratic equation You can't use my set I'm putting restrictions on your this thing so that you can think Okay Restrictions help us to think More Oh, you okay. Okay. Okay. I thought something else Sir, I think it should be the same Same Yeah, what about others? Minus a by m That's correct. Yes, cz minus a by m Now, how do I do this if I don't if you're not allowing me to use quadratic, how should I do this? So first what I'll do. I'll use the power of transformations So I'll do something such that this guy becomes y square is equal to 4x Okay I'm replacing my x with Minus x You have to do the same thing with this guy also Yes, sir Is a tangent to this You yeah, so I'll use the same condition, but now my c will be c a will be a but m will become a minus m Hence the condition c is equal to minus a by m Oh See, uh, one thing very important here is that if you transform equations, that means if you do the transformation to all the participants The condition doesn't change. Probably the equations are going to change the coordinates are going to change Yes, sir Now I'm giving you one more opportunity My equal to mx plus c is tangent to x square is equal to 4 a y Then what is the condition Without using quadratic same thing no quadratic using m by a C equal to a by m is equal to minus a by m Wait, wait, wait, wait, wait, wait Okay Minus a m square C is equal to Minus a m square Very good. C is equal to minus a m square How I'm using my initial condition only. So what I'm doing I'm taking the help of transformations to achieve my results What am I doing here Right, so you have to do the same activity here also. Yes, sir But right for the equation is not of the in slope intercept form So if I want to write it as slope intercept form, what will you write it as? Uh x minus c x by m minus Minus c by m Now the condition that this guy is a tangent to this So c is c by m a by m Yeah, I'm property writing it in capital alphabets. Now who's paying the rule of c here? C by m minus c by m minus c by m Who's paying the rule of a a A only Who's paying the rule of m one by m Oh Minus e by m is equal to a m that means c is equal to minus a m square. So Yes, sir So can you explain the last step again? See what I did was I applied transformations to these equations and brought it to a Y square is equal to 4a x form Yes, sir. Okay. Yeah. Yes, sir. I got All bridge squares stuff are in Yeah, yeah, fine. Fine. I know Okay, what are the conditions that this line is tangent to x square is equal to minus 4a y What is the condition? a m square Sir a m square sir properly What a m squared C equals a m square. I believe you. I'm going to put more questions on you. Yeah. Yeah, sure, sir. What is the condition that? Okay, let me give this as an objective Ah Y is equal to mx plus c is tangent to x minus x square is equal to 4 a m square Okay Okay The following option is correct option a c minus equal to k plus a m square option b c plus c is equal to a plus a m square option c c plus mh is equal to k minus a m square option b c minus mh is k minus a m square This is all Jay will ask you Jay being so nice and understanding Very good mx plus c Sir Option b b for ball Yes, sir Um Chal I will reserve my response here because if I say right or wrong people will rule out that Okay, Chal, Amla has said b What about others I will say w Sir not b, c I want to change c C Can we discuss it now No, sir How much time do we need? I am not getting an idea how to start Why? There is no original shift and all Uh huh But after that I am getting stuck sir See here What is the whole and whole purpose Whole and whole purpose is to make this as x square is equal to 4 a y right Yeah Yeah If I want to make this transformation What will I replace my x with And what will I replace my y with x plus h Capital x plus h So the moment you replace your x with x plus h and y with y plus k You know h and s will get cancelled k and k will get cancelled Right You realize similar thing has to be done With this boy You can't do step treatment with one Correct So what are you doing to do is you are going to Make this as y plus k as m x plus h plus c right Yeah Which means y is equal to mx Plus mx Okay, plus c minus k Yes sir Now this particular equation You are claiming that is a tangent to this guy Yeah When that can happen When c is equal to negative a m square How you learned that Yes sir, now we learned it Who is playing the role of capital c my dear C minus k So mx plus c minus k a will remain a Who is playing the role of m m only right So can I say c plus mh Will be equal to k minus a m square will be your condition Which is what Amla said later on Initially she was wrong but now she corrected it Option c Yes sir No but I am not happy, I will give you one more Yes sir So if y is equal to mx Which is tangent to Is tangent to x minus h whole square Is equal to minus 4 a y minus k Then Which of the following option Option mh is equal to k plus a m square Option b c minus mh is k plus a m square Option c c plus mh is k minus a m square Option d minus mh is k minus a m square So b Wait Amla wait I will just tell you Should a m square a a Because a m square Amla wait c minus k sorry a m square c plus mh equals k Sir a sir a c plus mh equals k plus a m square Option a is correct Want me to discuss it? Let's quickly do that The only way I can reduce this to x square is equal to minus 4 a y is replace your x with x plus h x plus h right So if you do this with x plus h You do this with y plus k You do the similar activity here also That means this line is actually y is equal to mx plus mh plus c minus k Yeah Now when is any line parallel Tangent to this when c square a m square is equal to a m square right So who is playing the role of c mh plus c minus k yeah a is a m square is m square So c plus mh will be equal to a plus a m square which is option a clearly So you are happy now sir? Yeah I am very happy Equation of tangents in various forms Now just now we learned that if we have a parabola y square is equal to 4 a x And this line is the tangent to this parabola then it can only happen when c is equal to a by b Yes sir I can also write the equation of a tangent as y is equal to mx plus a by b This type of equation of a tangent is called the slope form Why it is called the slope form Because if somebody says that there is a parabola y square is equal to let's say 8x Give me the equation of tangent to it having a given slope let's say slope is 3 So what is the equation of a tangent So we have to find this out having a slope of 3 The slope is known to you and you are going to find out the equation So what will be the equation So you can directly use this equation Whatever you have shown there y square equals y is equal to 3x plus 2 by 3 Jinx Okay But the biggest drawback with this equation is it is only valid for a standard standard may be only this case Okay It's a very restricted case However let me spend some time on this form also because it's useful in many type of questions So let me start with this question If If it's Y plus My plus equal to 0 and Lx plus My is If Lx plus My equal to 0 is a tangent so y square is equal to四a x okay Then Which of the following option is correct Option A LN is equal to AM square. Option B. LM is equal to AN square. Option C. AL is equal to MN square. Option D. AL is equal to NM square. A sir. What is the answer? A sir. A. A is correct. This is a very straightforward question, right? P is A sir. I just have to write this as Y is equal to minus L by MX minus N by M. Use the condition of tangency. C is equal to A by M. Yeah, yeah. Minus minus goes so N by M is equal to AM by L. N. Yeah. LM is equal to AM square. Next, if X by L plus Y by M equal to 1 touches Y square is equal to 4AX plus B. Then, which of the condition is correct? Option A. M square A plus L plus BL square is equal to 0. Option B. M square L plus B plus AL square is equal to 0. Option C. L square A plus M plus BL, BM square is equal to 0. Option D. L square M plus B plus AM square is equal to 0. Sir, for one minute, can you go to the previous question? Sure. It's the second one, B. A, wait. Okay. I'll wait. I'm not going to. Okay, okay. BM. Y equal. Sir, I made some mistake. AM square. Not AM square. This is Y. C is equal to A by M. Oh, sorry. Sorry. A by M. Wait. Got it. What is M? It's a two minutes more. M square L plus B equals L square A. Sir, M square L plus B plus L square A equals 0. So option B, right? B, B, B. Yes, sir. B is actually correct. So very simple. If you want to make it Y square is equal to 4AX replaced with X with X minus B. X minus B. Yeah. Here also. So X minus B by L plus Y by M equal to 1 is a tangent to Y square is equal to 4AX. So the condition is C should be equal to A by M, right? Yes, sir. So who's playing the role of C here? C. ML plus B by L. Let me write it in the proper format. So I'll write it as Y by M is equal to 1 minus X minus B by L. Yes, sir. So M minus X M by L plus BM by L, right? Yeah. So the role of X is in bed by these two guys, isn't it? Yeah. So M plus BM by L is equal to A. A will remain A. By? Minus M by L. Okay. So I can say here ML plus B by L is equal to AL by minus M. Yes, sir. AL square plus M square L plus B is equal to 0. Yes, sir. Which is option number B. Option if, not if, find the equation of common tangent, Y square is equal to 4AX and X square is equal to 4BY. Find the equation of the common tangent or tangents depending upon how much you can find. These two parabolas. So what is the maximum number of common tangents two parabolas can have? Two, right? Two parabolas. Yeah. You can have three also. No, you can have, you can have depending on the case. Let's say they touch each other at a common point, then one can be possible. They can be one cross also. So one, two, three, four can be possible. In case depends upon, three is possible definitely. Four we have to choose. Okay. Oh, okay. Wait, sir. I think. Only one answer is this. Don't worry. Oh, okay, sir. So is it LX plus MY plus M square A by L is equal to zero? There is L, my dear. There's only A and B in the question. Oh, okay. Oh, okay. Wait, wait. See, a lot of time you are taking this. If you are thinking that there's a tangent to this fellow, can I say, can I safely assume that the tangent is this? Yes, sir. Yeah. I am able to find M my job is done, right? Yeah. Now you're also claiming that this is tangent to this guy. I say C should be equal to minus AM square. Yeah. That means M is minus A by B cube root. Yes, sir. I got a weird answer. The answer is slightly weird. Yeah. Up next over here, your job is done. So your answer is Y is equal to minus A to the power one. Minus A by A to the power one. If you Y is equal to negative B to the power one. If I throw it with B to the power one third, one third Y equal to minus A to the power two. She has a different talk. That means it is A to the power one third X plus B to the power one third Y plus B to the power two third B to the power two equal to zero. Did you get the same thing? Yes, sir. Yes, sir. Yes, sir. Yes, sir. Yes, sir. Yes, sir. Yes, sir. Yes, sir. Yes, sir. If tangents to the parabola, Y square is equal to four AX. Theta equal to theta two with the positive direction of the X axis. That means anti clockwise with the X axis. Such that, such that cot theta one plus cot theta two is always a constant C. The locus of the point of intersection of these two tangent of the two tangents. Option A. A Y is equal to C. Option B. C Y is equal to A. Option B. Sorry. Option C. Option C. Y is equal to AC. Option D. Y is equal to A plus C. Any idea anyone? No, sir. Okay. So let me help you out with this. Again, let's say there's a point where you're drawing a tangent. Okay. Okay. This tangent makes an angle theta one. Another point you're drawing another tangent. And this angle is theta two. I have to find out what is the locus of the point of intersection of these two tangents. First of all, I'll assume that as if from point H comma K, I'm drawing a tangent to this parabola. You know that the tangent equation to a parabola Y square is equal to 4AX is this. Yes, sir. This is our standard case may Y square is equal to 4AX one. Okay. Now can I say H comma K will simplify this equation. So K is equal to MH plus A by M. Yeah. Correct. So can I write this as M square H minus M K plus A is equal to zero. Okay. This is clearly a quadratic in M. X square minus X square. Yeah. You can draw two tangents. So this will have a slope of M1 and let's say this has a slope of M2. M2 is the root of this quadratic. Till this point is clear. Yes, sir. Okay. What is M1? M1 is tan theta 1 and M2 is tan theta 2. Oh my God. Yeah. What is given to you here is 1 by M1 plus 1 by M2 is equal to C. That means M1 M2 M1 plus M2 by M1 M2 is equal to C. Yes or no? Yeah. What is the sum of the roots? Minus B by. Minus B by. Correct. So can I say it will be K by H? Yes, sir. And what is the product of the roots? A by H. A by H. So let us put over here. So ultimately I have to strike a relationship between K and H, right? So I can use this. So H1 by H1 by H1 gets cancelled. So K will be equal to AC. That means Y is equal to AC. That means option number 3 is correct. Oh yeah. So one more question sir. Again the same old question. If Y is equal to Mx plus C is a tangent to this parabola. Y is equal to Mx plus C is a tangent. Basically you know that Y is equal to Mx plus A by M is a tangent. Yes sir. What is the point of contact? What is the coordinate of the point of contact? One second sir. One second sir. 2Ax is equal to M squared X squared plus A squared by M squared. M squared X squared by 2Ax plus A squared by M squared 0. And 4x squared by 2M squared Ax. So I am getting a really long answer. The X coordinate itself is 2 plus M squared into under root M squared minus 4S squared by 2M squared. See if you want to find the point of contact it means you are trying to solve these two. Simultaneously satisfy those two. So I can say Mx plus A by M whole square should be equal to 4Ax. Yes sir. This should be a quadratic which is a perfect square. Why perfect square? Because it is only touching at one place isn't it? Yeah. So if you expand it you will realize you end up getting A squared X squared plus A squared M squared plus 2Ax and minus 4Ax. Yes sir. So if you give you M squared X plus A squared minus 2Ax equal to 0. Yeah, I got this. Which really means that you can write this as Mx minus A by M whole square. Yeah. That means X is equal to A by M squared. Yes sir. Now this point is having the X coordinate as A by M squared. The Y coordinate could be obtained by putting it in this. So Y is equal to Mx means A by M plus A by M. That means this will be 2A by M. If you really compare this with AT square comma 2AT you would realize that T here is 1 by M. Yeah. Very surprising. That means the parameter at any point is negative is reciprocal of the slope of the tangent at that point. Yes sir. So please remember this is also very very useful. Sir you know I got till M squared X squared minus 2Ax plus A squared M squared. Then I started applying quadratic formula. You can use the line equation. Yeah, yeah, yeah. Okay, so we'll stop here. Okay. And next class when you meet I'll speak something about the point form. This is the next class Ajay. In parabola entries I will talk about the point form of tangent. I'll talk about the normal. And we'll stop the parabola till summer vacation. Okay sir. I'll begin with ellipse. Second you wait sir. Thank you sir. Thank you sir.