 conic section okay conic section Conic section is a very very big chapter and it forms the major part of your coordinate geometry So coordinate geometry is comprised of normally six chapters. We say One is circles. Okay, I'll write them down Circles parabola ellipse hyperbola pair of straight lines and you have straight lines that is the six chapter that we have already covered Okay, so we don't have to worry about it. So these are the five chapters, which are normally covered under conic section now Pair of straight lines is not there for your class 11 So this will be something which will be doing exclusively after your after your exams are done Okay, so we'll start today with circles and Parabola, okay, they're very easy concepts. We can cover them in one classes one class now first of all Why is the name conic section given to these five geometrical figures? So let us understand that as an introduction. So let's have an introduction to conic section So introduction to conics Okay, now they're called conics because these structures or these geometrical figures are obtained by intersection of a right circular double cone With a plane now, what's the right circular double cone? Let me just show you just a second Yeah So this is a right circular double cone as you can see I have shown four figures Okay, and all the four figures you are seeing basically a double cone structure, right? Okay, that is called a right circular double cone Right. So a right circular double cone First, let me talk briefly about it Let's say I make a small diagram of a right circular double cone So this is a right circular double cone. Why it is called right circular. It is because the basis of these cones Make a right angle with the axis of this particular Right circular double cone. This is called the axis. Okay, so there are some things that you must know about them So this is axis these two Cones basically are called the naps Okay, they are called the naps of the right circular double cone. This meeting point is called the vertex Okay, and These two lines they are called the generators Okay, so these are called the generators Okay, so these two lines are called the generators Now the angle that a generator makes with the Axis this angle or this angle, whatever you want to call it this angle alpha is basically called as the semi vertical angle semi vertical angle Okay, now, how are these conics generated these conics are generated when they're a plane when a plane Intersects these right circular double code. Okay, when a plane intersects this right circular double cone It creates a Area of cross-section or you can say it creates a cross-section which can be that of a circle ellipse Parabola or a hyperbola. Okay, so when does it create a circle? Let us understand that so to understand that Let's understand the figure number one over here. So figure number one shows that when this plane cuts the right circular double cone one of the naps in such a way That the angle that the plane makes with the axis Which is normally called as beta in this case as you can see this is the angle beta So the angle which the plane makes with the axis is 90 degree That means beta is 90 degree then it will create a circle So this cross-section area that you see over here, it will be that of a circle Okay, even if Even if this plane passes through the vertex, it will create a point circle Okay, a circle of radius zero Now when the very same plane cuts one of the naps at an angle beta, which is less than 90 degree But still greater than the semi vertical angle as you can see Beta here is less than 90 degree But still greater than the semi vertical angle then this cross-section area will be that of an ellipse So figure number two shows the formation of an ellipse Due to the intersection of the plane and a right circular double cone third diagram shows when the plane makes The same angle with the axis as the semi vertical angle that means beta is equal to alpha in other words If you see this plane direction and the generated direction are parallel Okay, so you are slicing the right circular double cone by a plane in such a way that the angle that the plane makes is Parallel to that of the generator or the angle is equal to the semi vertical angle in that case It is going to cut this cross-section area, which is that of a parabola. Okay, the pronunciation is parabola Not parabola. Okay, I'm like Many of us said is a parabola. It is not a parabola. It is a parabola Okay, parabola is basically the path taken by an object when it is thrown under gravity. Okay So when you throw something towards somebody it takes a parabolic path, isn't it? That is why it's called parabola. It's a Greek alphabet a Greek word Now if the plane makes an angle which is lesser than alpha, but greater than equal to zero Then it will basically cut both the naps. So both the naps in this case will be cut as of now Only one of the naps were getting cut Now here both the naps will get cut and you can see here the cross-section generated would be called as a Hyperbola. It looks very similar to a parabola, but the structure is quite different. We'll see when we do the chapter So this is your fourth case Okay This is your fourth case right Now if you make a plane in such a way that it is passing through the vertex of the parabola Okay, then the cross-section area will be that of a pair of straight lines So that is a separate topic altogether which we cover under a degenerate conic. Okay What is the word degenerate degenerate conics are those conics where the focus lies on the directrix now What is this word focus? What is this directrix? I'm sure most of you would have heard of it But I'm assuming that you people don't know about it right now I'm just assuming that I'm starting this chapter very fresh and very new with you all So we'll talk about it when the right time comes. Okay So this is the reason why they are called for it now few things you need to remember about conics All the conics whether you talk about circle whether you talk about parabola whether you talk about ellipse Whether you talk about hyperbola whether you talk about pair of straight lines Yeah, condition of a pair of straight line pair of straight lines is formed as a degenerate conic Where the focus and the directrix basically focus lie on the directrix. Okay, that becomes a case of a degenerate conic Okay, I will talk about it. I did the right time comes. I Hi, Shashwath. I have not taken the attendance. Don't worry about the attendance. Worry about what I have missed That's the first thing I should you should ask This is not a school that your attendance will determine whether you are supposed to write the grammar Worry about what you have missed. Okay Attendance is no Maya Now don't go and tell your school to say servicing attendance is no Maya, so I'm not attending the classes. Okay. Anyways any conic the equation the equation of all conics is Basically a equation which looks like this Okay, this equation is called a General second-degree equation a general second-degree equation do not call it as a quadratic equation It's not a quadratic equation quadratic equation is in one variable. Okay. It's a general second-degree equation Okay, so it is like the skeleton of all the conics whether you study circle parabola ellipse hyperbola pair of straight lines all of them Overall structure overall skeleton would be that of a second-degree equation like that like like homo sapiens Whether you talk about man or whether you talk about woman whether you talk about a child We all share the same skeleton. Isn't it? Of course, there's some differences in the way our skeletons are right? For example for a man the the sodas will you broader right woman the girdle will be broader child Skeleton would be smaller. Okay, so overall all the conic sections will have the same kind of Equation here and there a b h g fc etc. Oh, yes child have more children have more bones very well pointed out Why there is a two in sometimes yeah, that's a very good question Harry here And so there are some to entertain over here. That's a question that comes every time Sir why to sir why not h only why 2g why not g only why and a 2f why not f only yeah Those are very valid questions. They have been written because of certain conveniences. I'll talk about it when when time comes Okay, so remember your question had here and I'll answer this in very short time now How do you distinguish then between different conics as I told you looking at the skeleton? How do you figure out that this was the skeleton of a man or this was a skeleton of a woman or this is a Skeleton of an ape or this is the skeleton of a monkey. How will you figure out? Of course, there are certain characteristics that will help us to figure out the differences in the same way There is certain relationship between a hb gf and c that will help us to figure out the differences between different conics Right, so if somebody comes and gives you a general second-degree equation and says hey What does this represent? Then how will you figure out whether it is a circle whether it is an ellipse Whether it is a parabola. There's a hyperbola. There's a pair of straight lines. How do you figure that out? Okay, so we'll talk about that initially but without any proof because as we know we are running under time crunch Okay, but I will definitely come back. It's my promise to you and we'll prove all of these later on Okay, so in order to know in order to distinguish Which Koenig is what or which equation represents what Koenig we first evaluate a term which we call as delta This delta is basically formed by a b c plus 2 f g h Minus a f square minus b g square minus c h square Okay, so you would be given the equation So you would be knowing your a hb g f c. Okay, so by using those you find out the value of this number delta How many people ask me sir? Why do you call delta? Because it can actually be written like a determinant. So most of you are already aware of the determinant We we had a good experiences with determinants chapter earlier on Okay, so if you expand this term, it is actually going to give you a determinant which is obtained by a hg hb f gfc Okay, now if you see There's an easy way to remember this Be this determinant is symmetric about the Leading diagonal so you can see h here h here g here g here f here f here So that's a way to remember it. Okay, so if you just remember that there's an hg and f coming over here The other elements you can automatically remember. Okay, so many people say sir I'm I'm I always forget this expression. So you can remember the determine now if this determinant happens to be or if this delta value happens to be Zero Okay, if this value comes out to be zero. Yes straight away zero Then this particular second-degree equation that I have written in front of you on the top of your screen Will actually be that of a pair of straight lines Maybe that of a pair of straight lines How why please do not ask me right now because there's a huge derivation involved. I've promised you I will come back to it. Okay. I'm not You know missing this out. Don't worry about it But if you're still more inquisitive you have all the videos with you or chronic section 42 videos are there Okay, watch them. You'll come to know your answer. Okay, you'll get your answer from there. Okay now If Delta is not a zero If Delta is not a zero and if H is zero That means there is no x y term as you can see H comes with x y term So if H is zero and if a is equal to b And if G square plus F square minus AC is greater than equal to zero So you must be thinking I go from where all he's getting all these things. Okay again Take it with a pinch of salt right now. Okay, I'll definitely not help you out later on. Okay, then this would represent a real circle Okay, real circle is a circle whose radius is greater than equal to zero. Okay, so there could be circles whose radius are imaginary So we'll not we'll not talk about them actually. So we had we are doing real Cartesian coordinate system So we'll not talk about non real circles. Okay If your Delta is not equal to zero and your x square is equal to a b Then that Koenig will represent a parabola. Okay, plural is parabolae or parabolas. Okay if Your Delta is not equal to zero and at square is less than a b Then it would represent an ellipse It would represent an ellipse Okay, and if Delta is not equal to zero and at square is greater than a b Then it would represent a hyperbola It would represent a hyperbola in addition to these two characteristics That means Delta is not equal to zero and x square is greater than a b in addition to it if it also has a plus b equal to zero then it would represent a Rectangular hyperbola. Okay, so a special case in the hyperbola is rectangular hyperbola, which we'll be taking up in our next class along with ellipse, okay now Whatever I have done so far Most of them I have not proven it because it will take time and I'll definitely do it later on and second thing I would like to highlight many people ask me so Do we need to know equation of a right circular double cone? Do we need to know the equation of a plane? Nothing. Okay, even though two 3d figures are intersecting to create a 2d figure We are not going to go into any of the 3d figures. Okay, it is just for you to understand why it is called a Koenig in Reality, you are not going to see a Koen Equation you are not going to see a plane equation in these chapters Okay, of course a plane equation is going to come but that is in your 3d chapter 3d geometry chapter not in 11 But in 12th, okay, so don't worry here. You have to see Koen equation or plane equation. No, we'll not talk about that Okay, anyways, so today my agenda is to start with circles Circles in Hindi we call it as ayat circles Okay, now, what's the circle circle the definition the locus definition the locus definition of a circle is it's a path traced By a point Point moving in a plane Moving in a plane in such a way that The distance of the point Distance of the point of the point From a fixed point From a fixed point is A constant Okay, now not a new definition for you And most of you would have heard of it, isn't it so this six point basically is the center of the circle and this constant Is what we call as the radius, isn't it now based on this locus definition Will be deriving the equation of a circle. So let's say this is your fixed point alpha comma beta okay, and and There's a fixed And there's a moving point. This is a fixed point. By the way, this is fixed Okay, and there's a moving point P. Let's say I call it as H comma K Which moves in such a way that it always maintains a constant distance from alpha comma beta. So the path trace by P will be that of a circle. So as you can see I have drawn a circle over here So this will be the path trace by it. So how would I get this equation? Very simple I can use the locus definition over here. Let me call this as C. So CP is equal to R So CP is nothing but under root of H minus alpha whole square K minus beta whole square is equal to R Okay, remove the under root sign. So H minus alpha whole square K minus beta whole square is equal to R square now generalize this Generalize this by putting H as X and K as Y and this will give you the equation of a circle Okay, which we call as the center Why did I change the color? Yeah, this is called the center radius form of the equation of a circle Center radius form of the equation of a circle Okay, so please make a note of this So when you know the center and when you know the radius This is how we find the equation of a circle and this is called the center radius form Okay, there are a lot many forms which we are going to come across Okay, so it's very important that we give some names to them so that when I'm referring to it You should know what I'm talking about okay, and If your center happens to be if your center happens to be zero zero Then the equation of a circle reduces to this Okay, this is actually named as a standard form of a circle Watch out the names guys if I'm going to use it tomorrow. You should be aware. What I'm talking about Standard form now. Why do we call it as a standard form? I'm sure you would have no heard of standard forms in state lines also, right? There was a little standard forms to point form right slope point form Slope intercept form intercept form normal form distance form. What were they there was standard forms, right? Now why we call them as standard form because they are used very frequently Something which is you know standard means people are using it, right? This is the standard Size of a of a table right because people are preferring to sit on these kind of tables only so standard forms are the most commonly used forms and the reason being commonly used is because they are the simplest form of that particular curve, right? They are very they are very simple to use. They are very less complicated Remember a Koenig can be a very, you know a big expression the equation can be having x square y square x y x Y constant everything would be there as we recently saw in the previous slide, right? But this form as you can see it has minimalistic thing Okay, and that's why it becomes very simple and so many of the properties and many of the concepts that we learn in this chapter We basically derive them on the standard form and then extend it to a complicated form Okay, so I normally call these standard forms like the frog in the laboratory, right? Normally, what do we do in order to study the human anatomy? We cut off we dissect the bisect or whatever you want to call it on a frog, right? Because frog is easy to handle right we can't do it on an actual human being Okay, so these standard this is the standard form. This is the center radius form. Okay now Please make a note of this we are going to go do a little bit of analysis on this Any question so far all right a small analysis a very very small analysis if you expand this if you expand this equation The center radius form equation you are going to get let me expand it here So this equation if you are going to expand it you are going to get x square y square Minus 2 alpha x minus 2 beta y plus alpha square plus beta square minus r square I'll put them in bracket because they're all constants. Okay Now if you see this carefully What is the structure of this equation the structure of this equation is like this x square plus y square plus 2 g x Plus 2 f y And it okay plus a constant because this is like a constant, isn't it now somebody was asking you earlier Why do we use 2g and 2s? Okay, it's the the reason because we use it is they come from squaring activity Later on when we do parabolic ellipse hyperbola There would be a squaring activity happening while you are finding their equation and whenever squaring activity Happens on a linear expression like that. It generates that 2 x y 2 g x 2 f y kind of a term Okay, you'll get more clarity when we do the other chapters like parabolic ellipse and hyperbola But as of now just to immediately satiate your question Why 2g because see 2 was coming here 2 was coming here So we normally put the 2 in the general equation on it. Okay, so that you don't have to divide by half and all Now this form is again named as general form Sir, how many names are there sir? Yeah, these all equations have different different name. This is called the general form Please note down this general form One important thing that you must note down is in the general form your coefficient of x square and y square are Essentially one Okay. Now just a quick comparison if you know if you remember we had done the general second-degree equation of a Conic, right? What was that is a x square d y square 2 h x y 2 g x 2 f y plus c equal to 0, right? The comparison if you do you would realize that in the general form we normally keep a b and a b as 1 Okay, there is no h term. You can see here. Remember. What did I tell you when I was giving those conditions? Just turn a page and check. What did I write for a circle? Delta is not zero. That is fine. There will be no h term so h is zero and Coefficient of x square and y square which is a and b. They will be equal. So that is what is happening over here That is what is happening over it. Okay, right now If somebody gives you the general form and they say okay convert it to a center radius form Okay, or in other words find the center and find the radius Then you you must be ready with the result So what we are going to do next is we are going to compare this equation and this equation Any question anybody? Again, don't ask those questions as of now There is a proper explanation to that delta and all why that figure comes from where that figure comes Okay, I will explain that to you Aditya It is a very lengthy process actually because you have to go through pair of straight lines Okay, it has a lot of significance. It has a lot of significance, but I'll come back to Delta again So let's not worry about Delta right now. Okay. Yes So I was just talking about that this is the general form of the equation and I'm not going to compare the The equation in the general form with the equation in the center radius form. Okay, so if you compare these two These two if you compare, let me write it over it compare Of course x square x square is same y square y square is same. So here 2g is like minus 2 alpha That means alpha is minus g Similarly minus 2 beta is 2f. So beta is minus f Okay, in other words Please note down the center is minus g minus f if the circle is given to you in the general form Okay, so you don't have to complete this square You don't have to complete the square in x and y to know where is the center Directly from the general form you can get your center, right? Similarly, if you compare the constant c with alpha square Plus beta square minus r square. You'll come to know that r square is alpha square plus beta square minus c Okay, and Alpha and beta themselves are minus g and minus f so you can write it as g square plus f square minus c Just take an under root Just take an under root. You will get the radius. So these two formulas should be here in your mind because they will be very very heavily used okay, now Again, let's go back couple of minutes when we started the class When I was giving you how to identify whether a Koenig is a circle there was one term which I gave you G square plus f square minus ac that should be greater than equal to zero Now I'm sure that would be making sense to you because if your a is one Okay, that means I have made my equation in such a way that a and b has been made one each That g square plus f square minus ac is actually g square plus f square minus c And it is very important for it to be positive. If not my radius will become non real Right, that is why it is very important for us to have that term greater than equal to zero Okay Ash is nothing very important. I just compared these two equations here. So I compared minus 2f with 2g So g comes out so alpha comes out to be minus g and I compared minus 2 beta with 2f So beta came out to be minus Cool, I yo see everybody's internet is dying and everybody's asking can Which part do you want and all I only explain this know which part you want? Which part you want me to explain the whole thing? Huh, I understand that you were thrown out, but which part you want? I cannot start the chapter again No, how say what I know you do what I have Which part? From x square plus y square Okay. Yeah, so basically I was saying I was telling the class that If you compare your Equation that you have got over here by expanding the center radius form with the general form You'll realize that your a and b are actually one in this case Okay, your h term is not there. So h is zero and of course the other terms remain the same Okay, and then I said let us compare these two equations to know how is your alpha comma beta, which is your center related to these parameters gf and c Is that fine? Do you want one any more explanation? Good enough. Okay All right We'll just see a simple question on this. We'll just see a simple question on this Let us say my question to you is find the Find the Center and radius of This is an example question center and radius of this circle 2x square plus 2 y square is equal to 3x minus 5 y plus 7 is Equal to 3x minus 5 y plus 7. I would like everybody to give it a try I would like all of you to give me the answer on the chat box Okay, so give me the center. Give me the radius and meanwhile, I will take the attendance Think Good aditya Okay Okay, and rock very good Okay, arithra All right All of you are giving me different answers. That means the concept has not been well taken. Okay. That's a failure on my part actually I take the blame on myself See first of all, there are some people who are directly using the formula Okay So, uh, they're saying, okay, this is your 2g term Correct. This is your 2f term Right, and they're saying minus g and minus f is 3 by 2 And minus 5 by 2 Okay, I think one of you have mentioned this out of the answer. Please let me tell you this is Why it is wrong because many of you didn't bother to listen to me This formula when you're using You have to have this one as the coefficient of x square and y square If there is no one there this formula will fail That's what j tests you. That's why what any question will test you on Okay So the first thing as kind of rightly said You need to convert it to the general form right now. This is not a general form idea students This is not the general form because there is two sitting over here Two I don't want So first divide by two So when you divide by two this will become your equation of the circle You know in the general form now. This is your general form Okay And on this you are allowed to use the formula right so this is your 2g This is your 2f Okay, please do not forget the sign and as you can see i'm engulfing the sign along with Oh, sorry 2f Okay, this is your c Okay, so your center as we all know is minus g minus f So minus g will be what minus g will be three by four if i'm not mistaken Minus f will be what minus five by four Okay, so those who have got center as three by four minus five by four Give yourself a pat on the back. You are absolutely correct What about the radius? Radius you have already discussed under root of g square plus f square minus c which is under root of Three by four the whole square minus five by four the whole square minus c is plus seven by two Okay, let's do some calculations over here. I think this is nine upon sixteen 25 upon 16 25 16 and this is 56 upon 16 correct Multiply with an eight. Okay, so that we can take the other thing. So it's the root of 90 by 16 root of 90 by 16 is three root ten by four Okay, three root ten by four units Always in your exam also Please write units over here. Okay It's not a dimensionless quantity. It has got some units But what is the unit in maths? We don't care about it. We care that influence okay, so uh unfortunately I think Only few of you have got this right patik has got it right very good patik guy three Anurag has got the center. Correct. Okay 45 by four. I think it's root 45 by eight not 45 by four other. So there is a small mistake there also. Okay Now going forward I would introduce you to one more form of the equation of a circle which is called the diametric form Diametric form So what is this form? This is basically a very special case when you are given with the diametrically opposite ends of a circle. Let's say I know the diametrically opposite ends of a circle as x1 y1 and x2 y2 How do I get the equation of a circle? So most of you would be thinking so what is the need to do a new form here? I can find the center that will give me the center I can find the distance between the two points half it that will give me the radius I will use the center radius form Agreed absolutely you can do that method as well. But there is another way to look at the things So I can say that this circle is actually the locus of a point again I'm relating it to a locus which moves in such a way That this angle is always a right angle. So this angle is always a right angle. Correct. So I can use the fact that The slope of ap and the slope of vp And the slope of vp Will be giving a product of minus one Isn't it Don't we know that pen two lines are perpendicular their product of slopes is negative one, right? So now as a locus question if you're looking at it, let's call this point as h comma k I hope all of you follow the proper mechanism of finding locus always assume the point to be h comma k Okay, till you become, you know You know master of the game then you can start using x and y directly Until you become master you use h comma k to be very very safe Yeah, so slope of ap will be what slope of ap will be k minus y one by h minus x one correct Slope of bp will be what k minus y two by h minus x two. Okay, and this product is minus one So let's take the denominator to the Right hand side So this will become minus h minus x one H minus x two and let's bring everything to one side. So h minus x one H minus x two plus k minus y one k minus y two equal to zero Okay, now let's generalize it So if you generalize it it becomes x minus x one Sorry x minus x two y minus y one y minus y two equal to zero So please make a note of this because you would need this directly to solve questions Okay, so please make a note of this concept Okay, so this is a Direct and you can say a simpler version Uh that you can use to get the equation of a circle provided provided, you know the diametrically opposite ends of the circle Now before I go to the other forms, uh, I will definitely do some questions with you just to make you Comfortable with the concepts. So can I go to the next slide? in case anybody wants me to Stay for a while so that you can copy do let me know Okay, let's take questions Let's take questions A very simple question for you all to begin with If this equation represents a circle What do you think are the values of p and q? Very simple straightforward question If this equation represents a circle, what do you think are the values of p and q? Please respond to it on your chat box Very good pun of Good good good Everybody should answer this. This is the simplest you can get Very good. I think everybody has got it, right? So what are the characteristics of a second degree equation to represent a circle? The characteristic is the first one coefficient of x square and y square should be same So here it is p here. It is three So p should be three undoubtedly Right for the reason a and b should be equal correct Second thing is there is no x y term in the equation of a circle Right, so if you don't want an x y term to come in this equation You have to keep your qs too, right because your h is zero in a circle, right? So straightforward p becomes three q becomes two. Is it okay any questions? Any questions any concerns? all right Let's move on to the next one Feel free to stop me in case you want to Get an explanation for any concept Next find the equation of a circle passing through the origin with three x plus y equal to 14 and two x plus five by equal to 18 as It's two diameters Okay, so they have given you that the circle is there which is passing through origin and the two lines mentioned to you on the question are basically acting like Two of the diameters of the circle. So what could be the possible equation of the circle? I'm putting the pole on Once you're done put your response on this on the pole I think two minutes is good enough for time for this Now these are indirect questions, right? So uh, they will never give you the center and the radius directly It's like you know serving you marks No, no teacher will serve you the marks for you to just grab and eat it So they are making your life a little bit miserable. Okay, so they want you to slog a bit To get the center and get the radius and get the center radius form from there on or and of course to convert it into a general form Okay, so I've got a response If anybody is not able to see the Pole running you can give a response in the chat box also Not an issue come on guys two minutes about to get over only three people could solve in two minutes Both nine sapphire, okay, I can max give you 30 more seconds five four three two one kill Okay, okay, no issues. No issues. I have registered your response. Okay Only 11 people could answer this question within two minutes almost 45 seconds. Okay, that's a slow slow speed Okay, is what stops you from answering this question? This is under a minute question, right? So you have been given that there is a circle. Okay, there is a circle And there are two lines which are behaving as the diameters. So their intersection will give you the center And not only that the circle is also passing through origin Okay, it doesn't have a center at origin. It is passing through origin Don't don't get confused between these two. Okay. So what I will do to get the center. I will simultaneously solve these two equations Okay Do a small thing multiply this with two multiply this with three and subtract muddy So when you subtract you get 15 minus two which is 13 y 54 minus 28 which is 26 So y is equal to two and if y is equal to two my dear friend x has to be four So this center is four comma two So the distance of the center from origin will give you the radius So radius is under root of four square plus two square which is under root of 20 correct and now Let us find out the equation because we know the center and we know the radius by using the center radius form And if you square it up, it'll become x square minus 8x plus 16 y square minus four y plus four is equal to 20 So I think four 16 and 20 will get cancelled. It'll give you x square plus y square minus 8x minus 4 y equal to zero I think this is option number c This has to be option number c. So people janta who has voted for c. Oh my god, c got the most votes also good Okay, so c is absolutely correct Was it difficult? Was it worth three minutes? Okay, let's move on to the next question Find the center of a circle which is passing through these three points. Okay So this is a case where there is a circle passing through three non-colinear points Guys remember there cannot be one circle passing through three collinear points. It can never happen You can never have a circle passing through three Or you cannot have a unique circle You cannot have a unique circle one circle passing through three collinear points Okay, so these three points must be non-colinear, which I'm sure it is You have to find the center of such a circle Uh, let's have around two and a half minutes for this question And the poll is on almost one and a half minutes two responses Last 30 seconds five four Okay, enough one Okay, all right Most of you have gone with option number b b for bangalore Now there are various ways to solve this question one is by scamming itself, right So you can figure out which of them Is equidistant from these three points. Okay, but that would be a lengthy way because you have to find so many distance formulas Okay, despite that there are many methods to solve this question So one way is we'll we'll first consider that let the equation of the circle itself be this Okay, so I will say that let the circle equation itself be this And since your circle is passing through these three points, so let me just make a quick small diagram for the same So let's say your circle is passing through these three points Okay They must all satisfy this given circle, correct So let's put four and five so when you put four and five you get a 16 plus 25 plus eight g Plus 10 y sorry 10 f plus c is equal to zero. That's your first equation Put three four so you'll get our nine plus 16 plus Six g eight f plus c equal to zero. That's your second equation Put five and two you get 25 plus four plus 10 g plus four f plus c is equal to zero Okay, that's your third equation So three equations three unknowns g f and c. Let's figure it out So first of all, let me put it in a very uh sober way That means let's combine the constants. So this will give you minus 41 if i'm not wrong Okay, six g plus eight f plus c will give you minus 25 And here 10 g plus four f plus c will give you minus 29 Okay, now solving. This is not a Herculean task. We can just take the difference of two of them. So let's say I take the difference of the first two I get I'm so sorry if I have written something Incorrect over here. This is Eight Yeah, so take the difference you get two g plus two f Is equal to minus 16. So first g plus f is equal to minus eight. That's your first equation Okay, in fact, that's your third equation Okay, take the difference of these two you get four g minus four f Is equal to minus four that means g minus f is equal to minus one. That's your fourth equation Sorry, this is fourth. This is fifth my bad Okay Now from these two if you add them you get g is equal to minus nine by two And if g is minus nine by two your f is going to be minus seven by two Correct In other words your center will be your center will be minus g minus f That is nothing but nine by two seven by two that gives you Option number a as your answer. Okay. Now. This is one way to do it Right another way to do it which you can always try it out is Assume that's your center is some alpha comma beta. Okay And assume these points are pqr So you can use that cp is equal to cq So cp is equal to cq. This will give you one equation And cp is equal to cr that will give you another equation So two equations two unknowns alpha and beta you can solve for it. That's another way Yet another way is Many people prefer doing this Many people say sir I will Do this I will find out The midpoint of This is q Yeah, I will find out the midpoint of pq Okay, I already know the slope of pq. So I'll find out the equation of this particular perpendicular bisector Similarly, I will find out the midpoint of qr and again I'll find out the perpendicular bisector and these two lines wherever they meet that will be the center You are most welcome to do it, but let me tell you it will cost you some time. Okay Constructing equations solving them definitely eats up your time. Okay. This is one more way which I will definitely talk about You know after I have done matrices and determinants chapter with you. So there is a direct formula also. Okay formula for equation of a circle equation of a circle Passing through Three non-colonial points But I don't want you to use this anywhere because number one you have not you're not very sure about determinants, right? You all have not Uh, you know study determinants officially. Okay, and secondly your school will not accept it. Okay Your school teachers won't accept it. You may lose marks there So there is a four by four determinant Which when equated to zero will give you the equation and that equation is x square plus y square x y one x one square y one square x one y one one x two square y two square x two y two one And x three square y three square x three y three one. So if you expand this determinant Okay, now many people say sir four by four determinant expanding won't it take a lot of time? No, actually because there is one one one in the fourth column When you learn determinants, actually, there are several shortcut ways for you to expand determinants This is just for you to note down right now Do not use this in school since your request from my side Let's go use math Karnay school. Right if you use it, you will lose it Okay, you lose marks. Okay So this four by four determinant equated to zero will give you the equation of the circle directly from there Whatever you want to find out you can find out whether it is the center whether it is the radius everything can be figured out from there Okay, you can see here only the first row is made up of variables Rest all will be made up of numbers because x one y one x two y two x three y three will all be numbers given to you Okay, so this will not be a difficult thing to expand We will talk and use this when we have learned determinants officially. Don't worry about it Is it fine any questions So i've shown you so many ways to get to the answer Your choice whatever method you want to use Can you explain the ending of the first method ending of the first method? Yes, of course, ashwath what I did was I put the points and I got these three equations. I simplified it I took the difference of the first two and the second two and I got these two equations which I have numbered as four and five Okay, sorry for this smudging over there. Then I solved it simultaneously to get my g and f We know that center of a circle Whose general equation is x square plus y square plus two gx plus two of five plus equal to zero is minus g minus f So this is your answer Could you show the formula at the end? Of course, yes, I will show you this is the formula at the end Anybody who is good with determinants or who already knows determinant. Okay can use this Okay, but only in our monthly test. Okay. Yes one more thing. I got reminded monthly test Even though it says full syllabus, I will only test you till wherever we cover before 31st of January Okay Okay, thank you So let's move on to a few more questions The center of the circles of radius five touching this line at one comma one is Read the question read the question very very carefully Basically, they say that there is a circle and there is a line Okay And this circle touches this line at one comma one Okay And this circle has a radius of five units Now you all can very well figure it out that there could be two such circles. Okay That is why in your options you can see there are two two options given for every There are two two answers given for every option, isn't it? So let's solve this Two and a half minutes will be good enough Two and a half minutes is good enough Time starts now Actually circle is a very big chapter because if I start doing circles only normally I take around three weeks Okay, three four classes will definitely be taken up And I'm sure your school is not going to wait that long so It has got almost 30 to 35 concepts Last 30 seconds Dear all last 30 seconds Okay five four three two one go Vote vote vote. We are all just nine of you have voted Okay All right, so as you can see these questions are not direct The way you get it in school these j questions are not direct. Okay So let's find out the center in this case. Let's say our center is at h comma k, right? Now first thing I can use is the fact that the distance of h comma k from this line Okay, so this line is three x plus four y minus seven equal to zero that is given to us as five units correct So instead of using distance formula of h comma k with one comma one use the distance of point from the line So the distance of the point from the line we know is three h plus four k minus seven mod by Under root of three square plus four square. Okay. This is given to us as five that means mod of three h Plus four k minus seven is equal to 25 That means three h plus four k minus seven is plus minus 25 That means two possibilities arise from here one is three h plus four k could be equal to 32 Or three h plus four k could be equal to minus 18 Okay, so these are the two possibilities Now I need one more equation So what equation I can use I can use the fact that The slope of let's say I call this point as center and I call this point as p The slope of cp should be negative reciprocal of the slope of this line because they are perpendicular Slope of this line is minus three by four negative reciprocal is four by three Correct. So slope of cp will be nothing but k minus one by h minus one is equal to four by three Let's do a cross multiplication So you end up getting uh four h minus three k is equal to four h minus three k is equal to one Correct. Now, let's solve these two So three h plus four k is equal to 32 And four h minus three k is equal to one Okay, you can do one more thing Why not test which equations Satisfy it. Okay, let's check does four comma five satisfied. Let's check Four five will give you 32. Absolutely. Correct. Yes, I know And 16 minus 15 gives you one. Yes. Four comma five satisfies it. Okay So obviously this is one of my answers But till I test the other one I cannot mark it because there is a none of these sitting over here. Be very very careful What if four comma five matches minus two minus three doesn't Then the entire option a will become wrong for us. Right then you have to mark none of these in that case Getting the point Anyways, I will now test with the other two guys. That is four three H Plus four k minus 18 and four h minus three k equal to one. So it does Minus two minus three satisfied. Let's check minus two gives you minus six minus 12 minus six minus 12 minus 18 Absolutely This minus eight plus nine is one. Absolutely. So this also satisfies So it has to be your option number a Okay, so here are certain things that you can take away from this question Many people who jump to use distance formula avoid using it Second thing, once you have got your equations, you need not solve it always. You can test which options match those equations Okay, save your time wherever you can any questions here any questions So we'll move on to the next question next question is Tangents drawn from one comma eight to this circle touch the circle at a and b Find the equation of the circum circle of triangle p a b Okay, so let me first draw a diagram for you so that everybody is Fine with understanding the question So from a point p you are drawing two tangents to this circle Okay, let's say this is your point p Which is one comma eight You're drawing two tangents to this circle, which does the circle at a and b Okay Now what are the questions it are asking you the question setter is asking you to get the equation of a circle Which circumscribe triangle p a b. So let me just draw a circle. Oh, wow nice So far more so far awesome. So what is the equation of this game? Uh, let's have a poll running for this Let's take two minutes Time starts now. Okay. Very good I tj 2008 question. Very good last 30 seconds five four three one Okay, you will be surprised to know here that this is basically based on one of the concepts you have already done in grade 10 Yes It can be solved by a grade 10 concept if you recall there used to be a chapter called construction of circles, right? Okay, and there we used to give a question that if a circle is known that means if you know a circle radius Sorry a circle center And you know a point external to it And you have to draw two tangents to this circle. How do you construct the two tangents? How do you make these two tangents? Oh, sorry Does anybody recall how how used to make this construction? anybody Yeah, all right equal length. So how would I know what lengths to draw? No, no, no, what is the construction process? Tell me Right. See we used to find midpoint of cnp, right used to take an arc like this Okay Correct used to bisect this used to find the midpoint Then from this midpoint used to make a circle like this, right? Do you remember? Good old days, huh Wherever these circles used to cut the blue circle used to connect p to that point that used to be that That used to be the tangent. Correct. This was what you had learned in class in So what do you realize? What is the moral of the story here? The moral of the story is This dotted circle has got p and c as the diametrically opposite ends And it also behaves as the circum circle for this triangle p you can call it as a and b Right. So the job is done. My dear friends, you have answered this question The center of this circle wherever it is And this p will be the diametrically opposite ends of this blue circle So if you know the diametrically opposite end of any circle Who can stop you from getting its equation? Let's do it So center is what center is three comma two. I hope nobody's making a mistake in finding the center at this stage Okay, don't ask how three comma two Right. So, you know one comma eight and three comma eight You can always write down the equation in the diametric form, which is x minus x one times y x minus x one times x minus x two Plus y minus y one times y minus y two equal to zero Don't write a one over here. There's a lot of people who write a one by mistake So it's x minus one x minus three y minus eight y minus two equal to zero And your equation will come out to be x square plus y square minus four x minus 10 y plus 19 equal to zero option number b It actually did not require two minutes also if you are aware of this idea Okay, and yes, I kept keep on saying this Well, one day if you Teach your kids, you say never forget geometry while doing coordinate geometry. Okay In fact, never forget geometry and trigonometry while doing coordinate geometry. Okay all right So with this I'm going to now move on To another question Next question is A circle touches the y-axis at the point zero comma four And cuts the axis in a chord Cuts the x-axis in a chord of length six units Find the radius of the circle Very easy question Put your response on the chat box once you're done And I have made a diagram for you for your convenience By the way, there can be four circles. I've just done it in the first quadrant Uh, sorry, they can be two circles one in the uh first quadrant other could be in the second quadrant as well But anyways, we need the radius right very good print out that was quick Yeah, this is not worth 30 more than 30 seconds Okay Correct Hariyaran correct aditya nice nice good Yeah, adrija also correct. Okay. So all of you could figure out that the center of this circle The center of this circle would be something like h comma four only right because This should be exactly 90 degrees. That means you have to go 90 degrees from the y-axis That means you'll be parallel to the x-axis. Okay And we know that if I drop a perpendicular This is going to be three three Correct and this distance is four Right, so anybody can figure out that the radius would be following a Pythagoras theorem So r square is equal to three square plus four square and very aptly said answer is five minutes Okay, no brainer question. I'm sure all of you would have got it. Okay Let's move on to one more next If the intercepts of a variable circle on the x-axis And y-axis are two units and four units respectively Find the locus of the center of this variable circle Okay, now let's understand this question before you start before you pick up your pen to solve it So there can be several circles Which can leave an intercept of two by the way, everybody should understand the meaning of intercept here So I would just take a small Fraction of your time to tell you about intercept. See, let's say There is a circle Which is positioned like this. Okay. This is your x-axis. This is your y-axis Let me tell you this length is what we call as the x-intercept Okay, let me name it ab. Okay. So ab is your x-intercept Okay, and this length let me call this as cd This length is called the y-intercept. Okay. Now Variable circle means there could be so many circles which could leave the same two intercepts of two and four respectively on the coordinate axis so if a circle Let's say if I can consider that the circle is actually moving Or changing its dimension And its center is moving also in such a way That the intercept that it is making on the x and the y-axis are respectively two and four. What is the path traced by the center? So it's a locus question And we all know that j loves locus Okay, so let's try to solve this question Sir, is that a circle? I don't know I don't know Figure out from the equation figure out from the equation Now you know how to identify from a second degree equation, whether it's a circle ellipse parabola hyperbola So let's try to identify. What is it? Okay, uh, I'm sure many of you are not even trying because you don't know how to proceed in this question So I'll do one thing. I'll just discuss a brief theory with you. First of all, okay If let's say I give you a generic form of a circle And this is a theory that you all must know also. Okay. I purposely did not cover it as a theory I was basically waiting for this problem to come up so that we can discuss it Are they just let me just take some time to explain the theory then you can you know attempt the question So, uh, how do you find out the x-intercept value and the y-intercept value when you know the equation of a circle in the general form Okay, so here Let us try to understand what is x-axis x-axis is y equal to zero line Isn't it x-axis is y equal to zero line. Correct. So if you simultaneously solve these two equations Okay, let's see. What do we get? So when you simultaneously solve this equation you get a equation like this Okay, now this equation will have roots Which will basically correspond to the x-coordinate of a and x-coordinate of b So x1 and x2 will be the roots of this quadratic Which will correspond to the abscissa of a and b right Now we all know that the x-intercept is mod x1 minus x2 Correct, which is actually nothing but under root of x1 minus x to the whole square Which is nothing but under root x1 plus x2 the whole square minus 4 x1 x2 Correct Now you must be thinking why am I converting it to all these strange forms it is because if you know a quadratic And you know its roots are x1 x2 you can easily find out the sum of the roots which is minus b by a And you can easily find out the product of the root which is c by a Okay, so it'll be this which is nothing but two times and the root g square minus c All of you should note down this result because it is going to be very very heavily used In your many problems of circles Okay, now Why I chose a way to solve it by using quadratic equation when all I could do is Use Pythagoras theorem. Of course, I forgot my own Rule that never forget geometry, but I wanted to I know everybody knows geometry So I wanted to I wanted you to know something extra. Okay So I could also solve this question by using the fact that if the center is at h comma k Let's say this is h comma k. This distance is mod k Okay, and your radius is Under root g square plus f square minus c So let's say I call this as m point which is the midpoint of ab So you could figure out bm length by using the Pythagoras theorem Which is g square plus f square minus c whole square under root whole square Minus k square. So that will give you g square minus c Okay Oh, by the way, uh k here is minus i'll write it down k here is actually Minus g minus f right so k here is actually mod of f Mod of minus f. So it's mod of minus f the whole square Why mod because it could be negative value also. That's why we normally keep it as positive. Okay So ab is double of bm Okay, by the way, this is bm square So ab is double of bm shall be to under root of g square minus c So both these approaches give you the same result. Okay So this is a very commonly used result. So I would request everybody to note this down Similarly, you could also figure out that cd, which is the y-intercept Is to under root of f square minus c Okay Now having known this result now you can proceed with your question Right, so those who wanted 30 seconds one minute to solve this question go for it But these two results should be there in your Uh, this result should be there in your formula list wherever you are maintaining it x-intercept And this is your y-intercept let me write it down here cd This is your y-intercept Sir, you forgot the spelling of intercept Hmm Back to the question my dear. All right. Aritra when you're solving this question You should always give your answer in terms of x and y. Okay, Hariharan Okay, Aritra very good Okay Anybody who is trying who's trying very hard, Aditya Pranav Pratik Rishabh Solaj Rian Guy three Arjun Abhinav Good good Aditya nice. Acha those who have got the equation. Can you also comment? What kind of a curve is that? Is it a straight line? Is it a circle? Is it a parabola? Is it a hyperbola? Is it a pair of straight lines? What do you think what kind of a curve is it? character Aditya Okay. Anyways, let's let's discuss this. So guys, I've already done all the hard work for you You just had to you know, see it off in style. Okay. So what is given to us is To under root g square minus c is given to you. That is two And to under root f square minus c is given to you and that is four right In other words, you have been given that g square minus c is one F square minus c is c is four Okay, now what do you want? You want the locus of the center? I want this guy locus that means this is my x guy and this is my y guy Okay So I want to get a relationship between g and f Right. So what is the role of a c c? I don't want c beta Right. So in order to remove c subtract the results, right? So you'll say f square minus g square is equal to three Okay, c is gone And what takes just to place your f with a y in fact a minus five, but this is square doesn't make much of a difference So there you go. This is the equation that you will get and here all Let's say I want to identify which curve is it So let me again recall with you This is a x square by square zero x y zero x Zero f minus three equal to zero. Okay. Now first delta test you will do a b c plus two f g h minus a f square minus b g square minus c s square What does it come out to be a b c a b c a b c a b c will be three Rest all of them. I think Sunya Sunya Sunya Sunya So this is not equal to zero Right not equal to zero means cannot be a pair of straight lines right Now what is the relationship between h square and a b? h square is zero h square is zero a b is minus one Hope I'm not making any mistake here So zero is more than minus one. That means h square is more than a b Which conic satisfies delta not zero and h square greater than a b go and refer to your First page of today's class. You will get the answer In big and bold. It's a hyperbola. Actually, it's a rectangular hyperbola because a plus b is zero also Remember, I gave you that in addition to h square greater than a b if it also satisfies a plus b equal to zero It is a rectangular hyperbola Right, we'll talk about hyperbola rectangular hyperbola when the right time comes Okay, is this fine any kasta any any any kasta adreja nodiar. It's a rectangular hyperbola. It's not a parabola All right, so moving on to the last part of today's chapter Today's circle chapter not today's class. Okay, the last part of today's Circle chapter is your parametric equation of a circle. Okay guys, let me tell you this chapter is huge Okay, after parametric equation normally I do The position of a point with respect to a circle then we talk about the intersection of a line with a circle Then from there we go to the concept of tangents normals tangent also is slope form point form parametric form Normal also two types of equations are there point form normal form Then we go to the concept of pair of tangents chord of contact The equation of a chord whose midpoint is known Then we do the length of the tangent then we do the concept of pole polar Then we do the concept of diameter intersection of two circles condition for orthogonality family of circles It's a huge chapter. It's a huge chapter, right Sir, I feel like crying now, sir So of course we'll cover those all things but after your school exams are over now What is the parametric form? See, let me start with a very basic equation of a circle Right, by the way, what was the name of this guy? What did we call this tarse? We should call it as Standard form right standard form. Okay, so I'll start with a standard form And the reason I start with a standard form because it is the most It is the simplest equation any circle can have Okay, no x term no y term just x square y square and constant minimalistic. Okay now Instead of directly relating x and y can I relate x and y through a parameter and say x is r cos theta and y is r sin theta Right where theta here is a parameter So if you eliminate this parameter, you will automatically end up getting this equation, right This activity you would have definitely done in trigonometry or class 10, right So try to eliminate theta from these two you will automatically get your equation x square plus y square is equal to r square Okay, by the way, when you write a relation directly between x and y they are basically referred as the Cartesian form Okay And of course it is the standard Cartesian form And when you refer to it in terms of a parameter, we call this as a parametric form Okay, both the equations are Representing the same circle. Don't don't get me wrong here. They're not two different circles same circle But one written in Cartesian other written in parametric form Now a million dollar question comes from students Sir, why do we need it? Right? I'm happy with this form Yeah, of course, many of you are happy many of you 90 percent. In fact, I would say 100 percent work Is done by the use of Cartesian form in school But not when you're solving j e main questions not when you're solving many of the j e advanced questions The main use of parametric form Is you to choose a point on that curve Some parametric form basically gives you an hint of how to choose a point on that curve So let's say this is my circle x square plus y square is equal to R square. Let me draw my coordinate axes also So if I want to choose a point on this curve, right, how will you choose a point? Let's say this is my x square plus y square is equal to four curve. Okay, how would you choose a point on this curve? Right? 99 percent of jenta will say sir. I will choose it as x 1 y 1 Okay, and throughout the problem you have to keep reminding yourself that x 1 y 1 should satisfy the curve So what do you have done? You have taken Uh, uh, two, you know, uh, you know, you can say difficulty You have brought a difficult two level difficulty in your problem solving one. You have introduced two unknowns x 1 y 1 And secondly, you have to keep this always in your mind while solving the question So it's an additional burden. No, let's say if you forget this or if you forgot that x 1 y 1 should also satisfy this Right, so it will create a lot of issues in solving the problem Instead a smart chap. He will do this Let me take the point to be to cos theta comma to sine theta But this guy is only dealing with one unknown. That is theta lesser the unknown simpler your life will be And he doesn't have to worry whether this will satisfy the circle because it is an identity Any theta you take baba it will satisfy it, is it? So working with less variables And hassle free approach is the advantage of parametric form Now people ask me sir. Why not to sine theta to cos theta? See guys, there can be millions and billions and trillions of parametric form possible Okay, it is not a A thing written in stone that for this circle, this will be the parametric form. No I can write hundreds of parametric forms But this is the most convenient one Right, I can show you a few more Let's say what if I write it like this x equal to t and y is equal to under root r square minus t square T is a parameter You tell me which is more convenient to look which is more convenient as for you to choose a point You will definitely say the yellow one because they're under root n all come no Right, and you don't like to work with irrational quantities Right. Yes. First one has physical significance also. I'll give you one more. I'll give you one more. What if I write x equal to Let me give you this. Let me give you this r by root 2 cos theta plus r by root 2 sine theta And y is equal to r by root 2 cos theta minus r by root 2 sine theta If you eliminate theta, trust me, you're going to get x square plus y square is equal to r square. You can try it out You can try it out But is this convenient? Let's say, yes, sir. You know complicated it. You have not you have not made my life simply complicated Right, so there can be so many parametric forms, but the preferred one is this guy So you'll find it in many of the books also. Okay, and here the relevance of theta also I will show you so basically if you have to choose a point here This angle actually is your parameter So if you change that you'll keep on getting different different points on the curve Okay, so parametric form is basically nothing but trying to connect the two variables via a third party And it's not a new concept to you remember we had done distance form in a line That is actually a parametric form of a line Are you getting my point? So instead of saying x and y are directly related to each other You relate x to something you relate y to something and they're related in such a way that automatically their relation will become the same It's like saying I'll give you a very funny example is like saying x and y are husbands and wives Right, let's say they are couples You're saying x is the father of a child theta and y is the mother of the child theta. They will automatically become husband wife Okay, right. So this is what we are doing actually in this case Is this fine? now If you have understood this I would like you to suggest The parametric form for this suggest A parametric form for this Suggest i'm not saying That there's only one parametric form possible suggest suggest a parametric Anybody done just right done. No need to give me the answer. Don't need to type it out because I know writing theta and all is not a Convenient thing. You have to write theta theta Aditya has actually written it also Aditya are you sure? So what would be the parametric form for x square plus y square is equal to nine? That means both the curves will have the same parametric form Same parametric forming the curve should be the same What I want to say aditya that means for you this One and three they don't have a significance. They don't have a meaning or they don't will not feature it Ah, hurry up and absolutely just one second guys something somebody's at the door. Just bear with me Yeah, sorry somebody was at the door. Yeah Yes, so absolutely correct hurry up and so what we are going to do is we are going to treat x minus 1 as 3 cos theta So your x will become 1 plus 3 cos theta Similarly y plus 3 you treat it as 3 sin theta So y will become minus 3 plus 3 sin theta. So this could be a preferred parametric form. So this could be a preferred parametric form Yes, absolutely. Yes, okay now When things are written in this way people find it very convenient Why 3 why 3 here? Okay, just treat this as analog x square plus y square is equal to 3 square How would you write the parametric form for this? You'll definitely say x is equal to 3 cos theta y will be equal to 3 sin theta Isn't it? And now this x is actually small x minus 1 and this y is small y plus 3 Now you got the reason why a 3 is featuring over here Why was an r featuring over here? You never asked me here. Why was an r featuring over here? Got it. Okay. Now if I give this question in a general form, let's say I take another example So just a parametric form for this circle x square plus y square minus 6x Minus 6x Minus 8 y minus 11 equal to 0. Yeah Now when I give this in the general form many people start faltering. That's why I want you to test I want you to guys to you know answer this one. So just me a parametric form for this Okay. See guys, you don't have to worry too much Okay, when the equation of the circle was in the center radius form, how did you manage this? The previous example was a classic case for this. You said alpha plus r cos theta And y was beta plus r sin theta, right? Now, you know that your alpha and beta are just going to become the same thing as now going to become minus g Plus r will become g square plus f square minus c cos theta And beta will become minus f under root of g square plus f square minus c sin theta. Isn't it The role change alpha is minus g beta is minus f r is under root of g square plus f square minus c Right. So why are we not taking so much of time? So here In this in this example, I can directly say x is equal to now minus g G here is minus 3. So minus g will be plus 3 Plus under root of g square plus f square. Now, let us calculate the radius first radius here would be under root of 3 square plus 4 square plus 11 Which I had chosen in such a way that it comes out to be a perfect square. So I think 20s Yeah 36 under root is 6. So This cos theta and y is equal to 4 plus 6 sin theta Okay, now if you try eliminating the theta From here, you will end up getting your this equation. So if you eliminate your theta, you will end up getting the given equation Okay, and vice versa question can also be asked Vice versa question can also be asked You may be given a parametric form and you may be asked to find out the cartesian form Or you may be asked to find out the center and radius of that circuit So be prepared for both types of questions Anybody having any challenge in understanding this Brief introduction about parametric form Okay, let's see it off with few questions and then we can go to our new topic which is the parabola Ah No, I don't think so Anurag. This is going to be coming in your school Then sir, why are you teaching us sir? because You may need it in your DPP's Okay, I'm going to just send you one DPP on circle so you can we may require it Okay Question is find the parametric form of this equation By the way, this is not a right way to frame it. You should say find a parametric form Because there can be many parametric forms Try this out everybody. We have just done a question a little while ago Just say done. You don't have to give me the answer Just say done We are going to discuss it straight away Five people saying done. We are going to discuss it Okay, first of all center center. Where is the center my dear friends center is at minus p by 2 minus p by 2 What's the radius radius is? Minus 0 right, which is actually p by root 2 if I'm not mistaken Right So your parametric form will be minus p by 2 plus p by root 2 cos theta And y will be minus p by 2 plus p by root 2 sin theta So this is one of the parametric forms possible Is it fine any questions any concerns? Oh, sorry. It is just yeah. Yeah, sorry. Yes Correct. Correct. You are correct. Yeah. Oh, no, no, no. It was it my worst Sorry, it's alpha plus r cos theta right alpha was minus p by 2. Yeah, this is correct Oh, you're confusing me Okay That industry can we go on to another question? Okay, find the center and let me add one more thing from my site radius also for this circle As you can see they have mentioned this to you in a Parametric form Best thing would be to get this equation only in Cartesian form. So our center and radius will be covered under that only So get the equation of this circle in Cartesian form right now. They have given to you in a parametric form Excellent excellent So guys and girls, I would like to bring this to your notice getting to a Cartesian form is as good as saying eliminate theta That is the core principle Whether it is a circle whether it's a parabola ellipse hyperbola or any curve in this world If they have given to you in a parametric form and they say get me the Cartesian form It is akin to saying or it is similar to saying get rid of theta So how will you get rid of theta? So from here I can say x plus 1 is 2 cos theta y minus 3 is 2 sin theta The best way to get rid of theta is square them and add them Correct. No, if you square them and add them you'll get this correct This is the equation of the circle in Cartesian form. Now you can get everything you want center becomes minus 1 3 Okay, this is your center Radius becomes two units Right everything is answered So with this we will give give a brief pause to this chapter. We'll come back again after your exams are done