 So today our agenda would be to finish off parabola from your school point of view Okay, so it is the second of our clinic section topics. We have already done circles last class. I think last to last class Yesterday was a session on mathematical reasoning and today will be completing off parabola. There is nothing much in it Okay, it's not a very big chapter at least some school point of view. Yes Some competitive point of view. There are many more concepts which will be taking up once your school exams are taken care of. Okay So parabola as I already Mentioned that it is a part of a clinic. So how is the parabola actually constructed? Let's say this is a right circular double cone Okay, this is a right circular double cone. It looks like an R glass or something like that not exactly an R glass But it's just two cones Which have been connected. Okay, so something like this. So this is called a right circular double code right circular Double cone Okay, and this is the axis of this right circular double cone So axis is basically a line Which passes through the vertex of this right circular double code here and it is perpendicular to the base of the code Okay Now if this right circular double cone is cut by a plane Okay, this right circular double cone is cut by a plane In such a way that this plane is parallel to This plane is parallel to I'm just making a diagram here Yeah, this plane is parallel to this line Okay, this line. So let me just show you This is parallel to this line By the way, these lines they are called the generators of the right circular double cone. Okay Okay, so these two are called the generators of the right circular double cone So if there is a plane Which chops off this right circular double cone in such a way that it is parallel to the generators Then the cross-section area that would be obtained. So let me choose a blue color pen to show you that cross-section area So this would be a cross-section area that you would be seeing this cross-section area is that Which is called the parabola. So we call this cross-section area as a parabola. Okay, so this is how a parabola is obtained Okay, so parabola as a conic is obtained when a right circular double cone is cut by a plane In such a way that the plane is parallel to the generator Okay Generator of this right circular double cone. Now many people ask me, sir Why do you call that as a generator? Because normally the generator If it rotates about this vertex. So if there's a rotation happening like this Okay, as you can see on my camera It is going to generate that right circular double cone. That is why it is called generator However for our syllabus point of view, we don't need to know about generators and all that is beyond our scope right now It is just to connect you to why parabola is actually called a conic Now circles that we did In the case of a circle the plane is such that It is cutting a right circular double cone In such a way that it is perpendicular to the axis So if it is perpendicular to the axis means this angle is a 90 degree Let me just draw it over here This angle is a 90 degree then it would leave a cross-section area of a circle So this cross-section area would be that of a circle Getting my point Okay, so circle is another conic When the plane cuts, by the way, these are called naps Okay, it should it is good to know the names of These things so these are called this cone and this cone. They are called naps. Let me write it on this side They are called naps of the cone Or naps of the right circular double cone Okay, so the cone on the top the cone on the bottom they are called naps Okay, so when a plane cuts one of the naps in such a way that it is perpendicular to the axis of the cone it creates a cross-section area of a circle Okay Now this is to understand why parabola is called a conic Let us try to understand Let us try to understand Let us try to understand what is the locus definition What is the locus definition Of a parabola Let's try to understand So as a 2d figure, how do you define a parabola? Okay, so please notice notice down everybody By the way, the same definition you will keep hearing from me in other conics as well in ellipse hyperbola, etc You will get to hear the same definition from me. Okay So all of you, please note this down A parabola is a path a path traced by by a point moving in a plane moving in a plane in such a way that It's distance It's distance from a fixed point Is equal to It's distance from A fixed line distance from a fixed line Okay By the way, have you done this topic in school nps raja jnagar? Have you taken up this topic in school? Hello, am i audible? Have you done this topic in school? Okay, you have done this topic in school. So you have already you have already, you know, you're only aware of this locus definition Okay, great So let me just draw this scenario So let us say there is a fixed point Okay, let me call it as s And let's say there is a fixed line Okay, let's call this line as d Okay And there is a point Let's say the point is our point p Okay, and this is a moving point Okay, so this point is a moving point Moving point So this is fixed Okay, this is fixed This is fixed They're not moving anywhere. Okay. They are fixed in the 2d space Now this point p, which is a moving point it moves in such a way that its distance from s Is equal to its distance from d in other words In other words sp Is equal to p m Okay, by the way, you should also know that sp by p m is actually called e, e for eccentricity Okay, so for a parabola the eccentricity value is 1 Okay, we'll talk about eccentricity also in some time So this point p is moving in such a way I'm just making some Dots for for me to construct the parabola Okay, so this parabola is moving in such a way that wherever it is Wherever it is its distance from S is same as its distance from p. So let me just join them with dotted structure. Okay, so this kind of a structure is basically same Okay, this structure is that of a parabola Now parabola is not a new figure for us because we keep seeing parabola the moment we throw something at somebody Right, most of you play badminton When you hit the shuttle cock The cock will follow a parabolic path, okay When you throw a ball in the in the air towards somebody that ball will take a parabolic path Okay, not only that the use of a parabola is also there in case of optics Okay, later on you will learn when we are doing this chapter too much more detail Is that if a ray of light If a ray of light comes parallel to the axis of the parabola, this is called the axis of a parabola We'll talk about it in some time Okay, this is called the axis of the parabola. So if there is a ray of light which comes like this Okay, it comes. Let's say something like this Then after hitting the parabolic reflector After hitting the parabolic reflector that will that ray of light will automatically pass through this point s Okay, later on we learn that this point s is called the focus So this particular property of a parabolic reflector is utilized in the field of medical sciences Where let's say somebody wants to treat Some problem in his you know organ. So normally what doctors do they'll put a parabolic mirror Such that that problem creating object is at the focus and they'll pass the laser beam Okay, and that laser beam after hitting the parabolic reflector will pass through that focus Thereby killing that you can say malicious object. Okay, so it is used in the field of Surgery for kidneys if you if let's say somebody gets a kidney stone and all The doctors try to kill that kidney stone by keeping that kidney stone at a at the focus of a parabolic reflector and then passing the laser beam Okay, of course the opposite of this phenomena could be used in our Headlamps of the car. Okay, so I'm sure most of you would have observed that in the car the headlamps are Parabolic in nature. Okay, we call them as parabolic projectors So what happens is let's say if you are driving and you want to see till far Okay, many a times the driver needs to see very far. Okay, so there are two beams Most of you have already noticed would have noticed it So if you have sat with next to the driver, you would have you would just do the up down of the beam Okay dip dip on dip off he does right so dip means he makes the rate dip Okay, so you can see something which is very close to the car But if he switches on the You can say the bulb which is at the focus Then the ray will travel far so that he can see far in the on the road what is going to come Right, so it also finds its you know application in that feed as well. So optics in the field of medical sciences parabolic reflectors Or parabola has a major role to play Okay, so what's the local definition of a parabola? How is it a tonic? And uh, you know, how does the structure look like everybody is fine with that? I will I will show this to you on the geojibra tool as well So let me show that to you on geojibra tool. Okay, so what I'm going to do is I'm going to just make a Parabola for you Let's say y square is equal to 4x We will see the we will see these equations a little later on in our discussion. Okay Now this tool can actually show you the focus So I'll show you the focus also so focus of this parabola So as you can see it has shown the focus by this alphabet a Okay, and this tool can also show you the directives. So directives of This okay, so as you can see directives also is seen. Okay Now if I take a point on the parabola, let's say here Okay, and Let me connect it to the focus a As you can see it has given a name g to it and g value is right now 3.03. Okay Now I'm going to drop a perpendicular From b on to the directrix Okay, and let's say I call this point as c And let me show you bc length bc length is i You can see i length is also 3.03. Okay. Now what I'm going to do is I'm going to do g divided by G divided by Okay, right starts value is 1 Now will this value change if I make this point b move? Let's try to figure that out Okay, so I'm moving this point b. All of you please Have your eyes glued to this value Is it changing Is it changing? No, right? It is still at 1. It is fixed at 1. So no no matter wherever you go on this parabola That ratio Of its distance from the fixed point a to its distance from the directrix is always a 1 Okay, so that is fixed to 1 Yes or no Okay So this is how the parabola basically Is you know defined as a locus Is it find any questions any concerns with respect to the definition? Now certain terms, let's understand certain terms which we need to understand related to parabola These terms were not used in circles. So please be careful Uh while listening to them because whatever terms I'm going to introduce to you They will be completely new to you Unless until you have done the chapter in school But the same set of terms will also be required in ellipse and hyperbola as well. Okay So let's go and identify what terms are we going to come across in parabola? So terms used in a parabola if we write it in white Terms in a parabola Let me make a diagram and keep it so that I can use it Uh normally to make a parabola I use a technique I make a oval shape structure like this Okay And I just erase half of this structure Not exactly. This is not exactly a parabola, but just to mimic it I am basically using this. Okay, it looks you know Aesthetically wise good Okay now We already have seen that this is called the Focus, okay So this point is the fixed point from where the distance is calculated of every point That point is called the focus. So this point is called the focus Okay, this line that is the fixed line which we were talking about in the locus definition That fixed line is called the direct fix Okay direct fix Okay, don't call it as dielectrics There was one. I think Candidate who came to us for an interview So he was constantly like I asked him to give a demo on on parabola So he made this line and he was calling it dielectric dielectric I mean dielectric is something that you have studied in your physics. Maybe okay. It is not dielectric. It is dielectrics. Okay All right Now the line which is basically passing through the focus and perpendicular to the direct fix that line is called The axis of the parabola Let me draw it Okay, so this line is called the This line is called the axis of the parabola. Let me write it in yellow Okay And the point where the axis By the way, this is a infinitely extending arm. Okay, so it just keeps on going on and on Okay, as you have already seen we have done this in our Functions chapter when we were talking about the quadratic polynomial Even quadratic polynomial graph is that of the parabola. Okay, so the point where the axis meets the Parabola that point is called the vertex of the parabola. This is called the vertex Okay, note it down So focus Vertex clear everybody Okay, if you connect any two points on the parabola, okay, let's say I connect These two points This is called a chord of the parabola. Okay, so this is called a chord So chord is nothing but a line segment connecting any two points on the parabola that is called a chord If the chord passes through the focus If a chord passes through the focus Something like this Then this chord will be called as the focal chord Okay, we will call this as a focal chord right A line segment which is connecting two points on the parabola and perpendicular to the You can say perpendicular to the Axis of the parabola. So let me draw a structure like that Yeah Okay, so a line connecting Two points on the parabola and perpendicular to the axis of the parabola That line is called the double ordinate. I'm sure you would have heard of this term double ordinate We call it as double ordinate because the extremities of this double ordinate will have the same Magnitude of the y coordinate. Of course, they will differ in sign But the y coordinate will be the same because it is Symmetrically the position is symmetric about the axis of the parabola Okay Now the most important term that you will keep hearing is lattice rectum. Lattice rectum is a double ordinate Which passes through the focus Okay, so this line if you see here This is a double ordinate which is passing through the focus and this is given a special name called lattice rectum Okay Lattice rectum. This is a latin word for side line lattice means side rectum means line So since this line is located at the side Okay, that's this and that's just a name given to it There's nothing very hard and fast about what is the logic behind calling it a side thing all of them could be side line So they just started calling it as a side line and that name has carried on Okay, so lattice rectum is nothing but our double ordinate Which is also a focal chord Double ordinate, which is a focal chord Which is also a focal chord You may call it as a focal chord, which is also a double ordinate means the same thing Okay, so it's a focal chord Come double ordinate. Is it fine? Later on you will learn that Lattice rectum Lattice rectum is the smallest Let me write it like this note Lattice rectum is the shortest or smallest focal chord Lattice rectum is the smallest or shortest focal chord shortest focal chord That means of all the focal chords that you can draw to this parabola the length of the lattice rectum Would be the shortest of all of the focal chord. Okay, lattice rectum is the shortest focal chord of the parabola Please note this out very very important Okay, is it fine any questions any concerns? So I think the terms I will be repeating them again focus Direct clicks so basically in the locus definition the fixed point that we had used in our definition that was focus The fixed line that we had used in the locus definition that is the direct clicks Oh, it's okay. It's okay. Yeah And if you connect any two points on the parabola that is called the chord If the chord passes through the focus that will be called as a focal chord a chord which is perpendicular to the axis of the parabola that is called a double ordinate So this line segment is called the double ordinate And if there is a double ordinate which passes through the focus Then that is called the focal that is called the lattice rectum of the parabola Is it fine And we have already seen here that lattice rectum is the shortest focal chord. We will prove it little later on. Okay, we will Okay, give a proof for this later on when we are doing this chapter from your je point of view or from your competitive exam point of view Okay Any questions any concerns anybody has please do let me know So axis is a line which divides the parabola into symmetrical halves Okay, is it fine any questions Is it fine? All right Now we will be talking about standard cases of parabola Okay, so let's take them. They are four standard cases of a parabola We will derive their equations and we will analyze those parabola with respect to the critical Points as well as critical equations. Okay, so let me go to the next slide. So we are now going to talk about standard Parabola or parabolas By the way plural of parabola is parabola or parabolas both are fine. Okay So there are four standard parabolas or parabola. First of all, many people ask me sir Which type of parabola we normally call as a standard parabola? Remember we had a standard circle Does anybody remember when I was doing circle chapter, I told you this is called a standard form of a circle Can you tell me on the chat box which type of circle was called standard form of a circle? Right, whose center was at the origin absolutely. Yes So in case of a parabola we call those parabola as standard parabola Whose vertex is that origin? Okay, vertex is that origin is that origin? And axis is Along one of the coordinate axes One of the coordinate axes. All right, so let me show you one of such cases Uh, meanwhile, let me fetch that diagram which I had already taken a screenshot. Yeah, okay. Let's say this is my Parabola, so what do I do? Let me show you the axis here. So let's say this is my axis of the parabola Okay, and this is my vertex. So what do I do here is I make my coordinate axis in such a way that x axis is along So this is my x axis x axis is along the axis of the parabola And y axis is like this Yeah, so this is my y axis Such that the vertex actually becomes origin Okay, so this is the case where my parabola would be called as a standard case of a parabola Okay Now in order to find the equation we need to fix this up because without fixing origin without Uh, I know fixing the reference axes. It is very difficult to write down the equation, right? Equation is what equation is nothing but It's a relationship between The x and the y coordinates of the point and x and y are nothing but directed distance from these reference axes So we have to put put our reference axes somewhere. So this is how I have chosen my reference axes So my x axis is along the axis of the parabola and y axis as you know It should be perpendicular to the x axis And the vertex is chosen as our origin point Okay Now one more thing I have to assume here in order to write down the equation of the parabola So I have assumed that I have assumed that This point coordinate is a comma zero where where a represents the distance Of the focus from the vertex Now, please mind you I have written here distance distance means something which is always positive So remember this and don't make mistake regarding, uh, you know the value of a ever while solving any question A is always a positive quantity. Please note that Okay, so when you refer to a a means the distance of the vertex from the focus Okay, and that distance is always a positive distance. So this is your a Okay, so that distance is always considered to be positive Never ever Take your a as negatives Okay Everything will go wrong while solving questions So y square is equal to 4x since many of you have already done this chapter in school You're already aware that that is the equation of this type of a parabola In that a is always a positive quantity Okay, however, we are going to derive that also officially in some time. Don't worry about it Now if this is a we also know that this point will become This point will become minus a comma zero Right. Why it is so because see this point. Let me call it as n n and s Right are such points whose midpoint is v Because as per the definition of a hyper parabola Any point on the parabola is equidistant from the focus and the direct x So vertex is also one of the points on the parabola. So this distance And this distance are equal And if this is a comma zero and this is origin Then this point had to be minus a comma zero Yes or no So as to say that now the direct x equation has now become x equal to minus a line Okay, so this line the direct x has now become x equal to minus a line. So this is your direct x equation This is your direct x equation Right Now having known the focus coordinate and having known the direct x equation Now the time has come that we use our Locus definition to get the equation of the parabola So what do we do we choose a generic point all of you here, you know how to find out the locus I know what you do you you choose the generic you choose a point and call that point the moving point is h comma k Okay, so this is a point which is moving And it is moving in such a way that this distance ps Is equal to p m. Let me call this point as m correct So the point p satisfies the fact that sp is equal to p m Okay, this is the Locus definition. So from locus definition of a parabola ps or sp is equal to p m correct Now what is sp? Let us write down sp sp would be under root of h minus a the whole square k minus zero the whole square okay And what is p m p m is the distance of h comma k from this line x equal to minus a line, isn't it? So x equal to minus a line. What is the distance of h comma k from there? So you will say h plus a mod By under root of one square Right that will become your distance p m yes or no Correct, so this you can use over here and write it as modulus of h plus Any questions in The framing of this locus condition, please you let me know Any issues Now let us try to simplify it. Let's square both the sides Let's square both the sides So if you square both the sides you get h minus a the whole square plus k square equal to h plus a the whole square So on simplification this will give you h square plus a square minus 2h Okay, this will give you h square plus a square plus 2h h square h square cancels off Right a square a square gets cancelled off Okay, so what we are left with is k square is equal to 4 a h Okay, so once we have struck a relationship between h and k we can generalize them By replacing our h with an x and k with a y So when we do that we end up getting something like this y square is equal to 4 a x Okay, so this becomes your Equation of this kind of a parabola. So remember this is one of the four standard cases So there will be four standard cases of parabola. Okay, that means for all of these four cases The vertex will be at origin And the axis of that parabola or those parabola will be along either the x axis or the y axis So this is one such case By the way, when we when we refer to it by a name, I normally call it as Rightward opening parabola. What do I call it as? Rightward opening parabola. However, there is no such I can say a term which is used in the books. That is something which I have point for, you know, my own reference So this is a standard rightward opening parabola So we will see a standard leftward opening parabola standard upward opening parabola Standward standard downward opening parabola. Okay, so first note this down everybody And if you have any questions do let me know done Any questions any concerns now for this parabola We will also derive the length of the latter sector because that is something which is going to be very very useful And a lot of questions will be asked around the latter sector concept. So let me just Yeah, so this is my coordinate axis x axis y axis So latter system as I already told you it's a double ordinate, which is also a focal chord Okay, it's a double ordinate, which is also a focal chord. Yeah So as you can see here That this green line is passing through the focus a comma zero Let me call the extreme ends as l and r just to resonate with lattice rectum l r Atter system. So I would request you all That for this parabola y square is equal to 4x. Please give me the length of l r What will be the length l r coming to me? Please find it out and give me a response on the chat box anybody By the way today again, I am sitting in one of the classrooms of center academy Most of you would not have got a chance to visit, but let me show you this is how Normally people are seated here Okay Okay, so it's a it's a closed classroom We are located in hsr layout And uh This is me and this is the blackboard and you have a class warrior Oh very good Most oh Satyam, are you sure? Okay, check it out. Check it out once again So it's very simple to derive. So since this line is perpendicular to the x axis On this line, the x coordinate is not going to change. Okay, so l x coordinate will also be same as the x coordinate of the focus Okay, so let us assume that uh, the Uh, coordinate of l is a comma k now. We already know that whatever we get for k We just have to change the sign of that to get the k to get the ordinate for r Okay, so at least I know that a is the abscissa for l So what I'm going to do is since this point l let me write it down here Since l satisfies The equation y square is equal to 4 ax I can just put a comma k A comma k in this so k square is equal to 4 a a That means k square is equal to 4 a square So k is plus minus 2 a correct So this is going to be 2 a Actually, why not minus 2 a? Why did why did I choose 2 a here? Even though I got two answers Why not minus 2 a Because I already told you that a is positive So as you can see this point is located above the x axis So its ordinate will be positive right over right. That is why 2 a for this So your r automatically becomes a comma minus 2 a Okay, anybody can figure it out who has got the basic knowledge of distance formula that the distance between an l and r will become 4 a units. Okay, so please note this down and keep this in your mind Now many people ask me sir if the parabola tilts or it becomes word upward opening left forward opening downward opening, etc Does the length of the lattice rectum change? No, the length of the lattice rectum is not going to change irrespective of how you have you are orienting the same parabola So the distance between the vertex and the focus If you do four times that that will always be the length of the lattice rectum So I would like you to write this down Length of the lattice rectum Let me write it as a note Length of the lattice rectum Is always four times Is always four times the distance between Distance between the vertex and the The vertex and the focus Okay, this is going to be applicable to any parabola any parabola no matter how ever it is oriented The fact this fact that length of the lattice rectum is four times the distance between the vertex and the focus That always persists that always stands Is it fine Arvind? Any questions? Oh, no problem. No problem. Okay Okay, so without much ado, let me call this as the first of the standard cases Okay, without much ado, we'll go to the second standard case where our parabola is A standard case that means its vertex is at the origin And its axis is along the One of the coordinate axes, but it is opening leftward. Okay, so let us take that as well I will do some I will sketch some diagrams over here Okay, just for construction. I am doing all these shortcuts Because freehand drawing doesn't come come out very well. That's fine Okay, but please do not think that Parabola is half of an ellipse or something like that. Okay, that is not there's no relation like that Okay, so this is your vertex origin This is your x axis y axis And just to show the directrix. This is your directrix Yeah, this is your directrix. Okay, directrix is also a line. So let us put arrows And this is your okay now here as I told you this distance is a The distance between the vertex and the focus is a So we don't write it as a comma zero. We write it as minus a comma zero because we want to use the fact that a is positive Okay, because many people ask me, sir. Why don't you call that as a comma zero? Anyways, it's a point on the x axis Agreed it's a point on the x axis, but the fact that we are consistent with the fact that a is positive A basically represents the distance between the focus and the vertex So that is taking to be positive quantity always. Please make no mistake about it in whatever problems or whatever Exercise questions that you are going to get never ever My mistake also take a value as negative Okay, your entire problem will go for a toss Okay, so let's not make any mistake about that going forward Now since this is minus a comma zero and this is origin then this point Let me for the time being call it as n that becomes a comma zero In other words, the directrix now equation will be x equal to a Now you can derive the equation like the same way as how I derived for the first standard case But I'm not going to do that because I'm already aware of a graphical transformation Which I had learned in our bridge course What are the graphical transformation that I can use here to get this equation in a faster way? I can use the fact that This is nothing, but it is a mirror image of the first standard case about the y axis So if you're reflecting any graph about the y axis What gets changed in the equation of that particular curve? The sine of x absolutely. So without much I do I can say that for this case of a parabola My equation will become y square is equal to minus 4x Okay, and let me also remind you here that the length of the lattice system Is not going to change the reason being as I told you you have just You know taken the mirror image of that about the y axis dimension wise nothing is getting affected That means the distance between the vertex and the focus is still a So later symptom will be four times that distance. So that will be four a units Okay, of course, please note down the critical points here minus a comma zero is your focus zero zero is the vertex x equal to a will be your directrix y equal to zero is your axis Okay, so all these things should be I mean Let's let's make an image of it in our mind so that whenever we are retrieving this data We can just close our eyes and think of that parabola and everything we can produce from that idea Okay, is it fine any questions? Okay, so let's go to the third case without much I do In the third case, we will make a parabola whose vertex will be at the origin Okay, and the axis will be along the y axis In short, it would be an upward opening parabola So let me make a quick diagram Okay, and I'll just erase this part This is your x-axis. This is your y-axis the vertex as I already discussed. I will keep it as origin And this distance is a as I already know So I will call this point to be zero comma a because now it is on the y-axis Right And since this is a zero comma a and this is origin That means my directrix should be passing through zero comma. I am sorry Yeah, my directrix should be passing through Zero comma minus a okay, so this is my blue line is my directrix line So this point here is Zero comma minus a okay, so you can call this line to be you can call this line to be y equal to Minus a line. Okay, so please note this now now Using the locus definition. Let us try to get the equation of such kind of a parabola as well So let's say p is our moving point, but it moves in such a way that the distance sp Is same as the distance p m Okay, so using this definition. Let us try to derive the equation of this standard case as well So everybody do it and do let me know on the chat box. What do you think is the equation for this parabola the upward opening parabola I'm waiting for your response Neel has already done it before. Okay, good Anybody else? By the way, many of you would have already started working on your school projects I hope you know that in cbsc as well as in isc. You have to submit projects If you need any kind of a help or an idea from my side Okay, please stay in touch Okay, I will so in 12th also you have to submit projects I think it's worth 20 marks. I believe Okay, most likely you'll work it over the holidays. Okay Very good. Satyam. Okay. So without much ado, let's figure it out So sp sp distance will be nothing but under root of h minus zero whole square k minus a whole square, right Equal to p m Now what is p m distance? So basically this is the line y plus a equal to zero So p m distance will be mod of k plus a by under root of one square, which is actually mod of k plus a correct So use it over here mod of k plus a Fine, let's square both the sides So when you square both the sides, you will end up getting h square k minus a the whole square equal to k plus a the whole square Okay, so this will be h square k square plus a square minus two a k Okay, two a k so k square k square gone a square a square gone So what you will end up getting is h square is equal to four a k Okay, so when you take this guy minus two a k to the other side You are going to get four a k. Yes or no Now generalize it. So once we have reached this stage, we can generalize it So when you generalize it, your expression will become x square is equal to four a y Absolutely, right those who answered with x square is equal to four a y Exactly that is the equation of the parabola. Okay And as I already told you this before and I'm repeating it again. It is just that You have just let me ask you this question If you remember the first case of the parabola, which was your right word opening parabola Can you tell me what do you do to that so that you get this parabola? What graphical transformation you are applying to that first case That it results into this case. Can anybody tell me that? Okay. Now one answer that I'm getting from everybody is Yes, Satyam has given one of the answers which I was looking for. Yes You're obviously rotating the graph 90 degree anti clockwise, but what actually you're doing You are reflecting every point on that particular curve about y equal to x line Okay Right. So see what has happened. I'm I'm showing you a very small, you know, version of it This was your first parabola, right? Let's say this is our line y equal to x Okay, so what are you doing? You are taking this as a mirror. So think as if this is a mirror Okay, so you are reflecting this graph This the white parabola about this mirror. So see how it will look like This part is going to look like this By the way, this is a mirror silvered from both the sides. Okay, so this is going to look like this This part is going to look like this And this part is going to look like this Okay Right. So it became our so from y square is equal to four ax It became our yellow parabola, which is x square is equal to four a y. Okay And if you would recall, I would I had also told you a graphical transformation for this scenario as well So whenever you are reflecting any curve about y equal to x line In the equation of the curve, you just swap the positions of x and y That means wherever there is an x put a y and wherever there is a y put an x Okay, so exactly the same as happened with in our case here as well So in my y square is equal to four ax y became x and x became y So as a result, you've got x square is equal to four a y Right. So this further justifies that whatever transformations we learn In our bridge course, they definitely work. They have to work. They are actually tried and tested theories. Okay Is it fine? Any questions? Okay, so I was about to say length of the latter segment of the latter system still remains for a unix. It's not going to change Is it fine? Any questions any concerns? So what I want you to take away from this entire scenario is the mirror image of not mirror image. Sorry It's the image of this in your mind. Sorry I got into that mirror image concept so deep that I was using better image You just have to make an image of this in your mind so that Whenever asked you can replicate everything. So where is the focus? Where is the vertex? What is the equation of the direct mix? What is the length of the latter spectrum? What is the equation of the axis etc should be immediately coming to your mind the moment you imagine this Okay So let's take the fourth case, which is our last case So this was the upward opening parabola now. We'll go to the downward opening parabola. So the fourth case the fourth of the standard cases So in the fourth case In the fourth case Our parabola is going to open downwards So let me just make an image here So in this case also my vertex is at origin This is my vertex Okay, focus will be at a distance a here Okay, so this point will become zero comma minus a And directrix will be cutting the axis. Oh, sorry Yeah So the directrix will be cutting the axis at this point. Let's say I call it as n Which is zero comma a and hence this equation of the directrix will be y equal to a Okay Now without much ado, we can find out the equation of this parabola as well because It is just the mirror image of the third case about the x-axis So when you are taking a mirror image of any curve about the x-axis, what do we normally do? What do we normally do? Write it out on the chat box Fast fast fast We change the sign off Yes, we invert the sign of y exactly so it will become x square is equal to minus four a y Okay, and let me tell you the length of the latter system is still going to be four a only It's not going to get affected because dimensionally nothing is changing dimensionally nothing is changing Is it fine any questions any concerns please immediately highlight Okay Now based on this locus definition and based on whatever four cases they have taken let us start solving few questions Okay, so let's begin with some questions If everything is copied do let me know I would like to go to the Next slide for a question. Okay, let's take a question All right, so this is a question which is based on your locus understanding of the parabola Find the equation of the parabola whose focus is that negative one comma negative two and directrix is the straight line x minus two y plus three equal to zero Once you're done, you can either say I've done or you can give me a response on the chat box either way you're convenient Neel is done very good. So like the Parabola is going to appear like this So in this case, you will see that it becomes a case where your Axis of the parabola is not Along the x-axis in fact, it is not even parallel to the coordinate axis. It is basically inclined at a certain angle Okay, so this is your axis Okay Anyways, we just have to follow our locus definition of the parabola to get the answer here Very good. Satyam Let's say the locus Of this point p creates this parabola Okay, so we have already seen that as for the locus definition sp distance should be equal to P m Correct So sp should be equal to p m. So this is a locus definition for any parabola Okay, so what is sp distance under root of H plus one the whole square k plus two the whole square Okay equal to p m now. What are the distance of h comma k from this line? So you'll say h minus two k plus three mod By under root of the square of the coefficients of x and y At the point Yes or no Yes or no Okay, so let's try to simplify this to a certain extent You can square both the sides first of all And When you square the denominator, it actually becomes a five. So I'm bringing the five to the left hand side Okay, and I'm simplifying it. But when I simplify it, you already know the my way of simplifying it I don't write everything. I just focus on the types of terms that we will be getting For example, we'll be getting h square. We'll be getting k square. We'll be getting h k terms x terms Sorry, h terms k terms constant terms, right? So I'll be using that So I'll be using that direct method to simplify it. So this will be five h square and minus h square will give you four x square So you will have five k square from here and this will be four k square. So that will leave me with a k square This will be giving you minus four h k. So on this side, if it comes, it becomes plus four h k And you will end up getting 10 h from here and six h from here. So that will be four h This will be 20 k and from here you will end up getting minus 12 k. So that will give you 32 k Okay, constant here will be 25. Here it will be nine Okay, so this will give you plus 16 Okay, so please let me know if you have if you have got the same or please also let me know if if you think I missed out on anything Okay, oh them got the same very well done. So here we will just generalize it. So after getting this we will generalize it We will replace our h with an x and k with a y So there you go. This becomes our equation of the of the parabola Now the look in the field of this parabola equation basically tells you that The parabola that you may get may not be only having like, you know, simple terms like x square 4 a y y square 4 a x etc. It could be as complicated like this That means you could have terms like x square y square x y x y constant also occurring in the parabola Okay, absolutely right arvind. I think your answer also matches with mine Is it fine any questions any concerns anybody has please do let me know Please do let me know Okay, let's see whether there is any more questions related to them We'll come back to these questions. They're all Okay, we'll come back to these questions. So first I would like you to do one basic question here Fine Number one Vertex Number two focus number three equation of directrix number four equation of axis number five Length of lateral sector Okay for for These two cases. So I'll just give you two cases to work on y square is equal to let's say minus uh x that's first case And x square is equal to 16 y Okay, so they are both standard cases. They should not take much of your time And when you're doing it, please give me your answer in one go Okay, so don't give me vertex and then press enter. Maybe you can do one thing Uh, you can answer the first one a first as of now I'm requesting everybody to give the answers to the a part of the question So put one two three four and put all the No answer that you feel and then press enter. Okay, so that I get all your answers in one go Please try this out. Very simple version I think next week we should be able to complete ellipse and hyperbola as well from your school point of view and Then we'll get back to our normal routine classes of one day a week class And I think we'll only have a bit of trigonometric Part to be covered trigonometric equation property of triangles And yes, we will also touch upon binomial theorem even though it is not coming for your school exams But that's a very important concept in general Anything else that I'm missing on we did derivatives. We did probability Yeah, so I don't I don't think so anything else is left Yes, uh, anybody with the answers to the first A part of the question So for a Okay, Vishal very good Uh, what about length of lattice system that you forgot I would eat Okay Anybody else for a Okay, Satyam Harshita very good Okay, let's discuss this out the first and the foremost thing that you need to do is you need to ask yourself This standard case Is which of the four That we have discussed it is the second one So this resembles y square is equal to minus 4x Please do not compare this with please do not compare this with y square is equal to 4x The reason being if you compare this with y square is equal to 4x Your a will become negative one by four which I said Is a complete no no, please don't write your a as negative because it is representative of the distance between the vertex and the focus Okay, so we can compare it with this. So when you compare both of them, you realize minus 4a is minus one So a is one four. Okay, please note We always take a is positive always. Okay, never ever take a as negative Now just close your eyes And just imagine this parabola in your mind the left word opening parabola Okay, and start answering these questions. So where is the vertex for such a parabola origin? Where is the focus of such a parabola minus a comma zero? So this will be minus a comma zero correct What is the equation of the directrix again close your eyes equation of the directrix is x equal to a Correct. So you'll get x equal to one by four right what is the Equation of the axis equation of the axis is your actually the axis is your x axis So x axis equation is written as y equal to zero What is the length of the lattice rectum length of the lattice rectum is 4a units 4a unit means one unit Okay, so if you have got all the five give yourself a pat on the back absolutely, right? Okay Anyway, these are the questions which you can get in your school also one marker. So let's quickly do the b part as well Okay, b part anybody is ready with their answers. Do let me know Okay Okay, Nandita Satyam Harshita very good. Very good. Very good. Arvind. Okay. Okay Arya, where is your answer? You have been keeping quite all this while Rohan smithy Venkat Vihan Tanvi Not sure why This is the easiest you can get. Okay. So x square with what what a standard case will you compare this with? Obviously the upward opening the third case. So if you compare this equation With this equation x square is equal to 4 a y You automatically get 4a as 16 So a becomes 4 Yes or no Now close your eyes and just imagine A upward opening parabola Okay, everything related to an upward opening parabola Okay, so where is the vertex? 00 origin Okay Correct Where is the focus focus is 0 comma a so 0 comma 4 is the focus What is the equation of the direct flex y equal to minus a correct What is the equation of the axis axis is actually your y axis So x equal to zero is your equation of the axis What is like a symptom length 4a 4a will be 16 units Then what is the issue in this? Correct Arya now you're sure how to do it Okay Irwin so later symptom is 4a Okay Do you do you want me to give you one more problem or is this fine? I mean these are all easy stuff. That is why I'm not giving you Okay, it's fine. Okay, so now we are moving on to the shifted form of the parabola So we have already completed our standard form. So there are four standard cases Of a parabola. Now we are going to move towards shifted form of parabola By the way, shifted form is also called the generalized form not the general form. Don't get confused. It is called the generalized form so now Let me make one of the standard cases the first one. Okay, so let's say this is our okay. I already have an image So let me just borrow it from there So, let us say this is the This is the Let me make my coordinate axes also. This is my y axis x axis I've already made Okay, so this is my This is my y square is equal to 4ax parabola. Okay Now I would like you to tell me If I shift this parabola Like this Let me erase this. We don't need it. Okay, let me let it be okay. So if I shift my parabola like this, this is my x axis This is my Let's say y axis Okay, this is my x axis. This is my y axis Okay, this is the origin Oh, sorry. This is the origin So what takes has now come here Okay, but when I'm shifting it I'm shifting in this way that The new axis of the parabola Is still parallel to the old axis. That means this is still parallel to The old axis and now the vertex has come to alpha comma beta Okay, the vertex has come now to alpha comma beta Okay, this green line is the directrix line. I have shifted the directrix also along with the curve Now tell me what is the equation of this parabola now? Who will tell me what is the equation of this parabola now if you want you can write it down on the chat box as well Tell me Okay, so Vishal says Okay, Vishal, are you sure? Have you swapped the positions of something? I feel so Okay Now let me take you back to your shifting of origin concept See when your origin was shifted to some point h comma k Okay How does a equation of a curve get changed because of that? How does this equation get changed because of that? Tell me Let's say f x comma y is a curve. How does the curve equation get changed? What do you do to you know get the new equation? See you already know that your small x is capital x plus h Your small y is capital y plus k, correct? Yes or no? Yes or no? So can I say the same equation will now become capital x comma y equal to zero where Where in fact you're replacing your small x with capital x plus h And small y with capital So let me write it like this better to be Yeah, so you're replacing your small x with capital x plus h Small y to be capital y plus k Correct Yes or no, this is your new equation correct Yes or no Do you remember this? This was your old equation. This is your new equation Correct Now you tell me everybody think carefully before answering it. Don't be in a rush to answer this In our situation, where has the origin got shifted to? When you are going from this situation to this situation How have you shifted your origin so that this curve becomes the second curve? What is I mean indirectly I want to ask you what is h and k here? No, Arshitha I knew people are going to make that mistake No, Arnav Right Satyam Your h and k is actually minus alpha and minus beta c all of you please notice this once again see You want your curve to go up and right? Let us say assuming that your curve is going up and right it could go in anywhere You know as per the requirement of the question if you want the curve to go up Origin should come down. No because the curve person is not moving Are you getting the point this red curve this red parabola it is fixed. It is not moving who's moving origin is moving, right? So if you want the curve to appear to have gone up origin should have actually gone down Yes or no Correct and if you want the curve to appear to move right origin should have actually gone left Correct. So if your vertex has gone to let's say Alpha beta your origin should actually in order to you know achieve that Your origin should have gone in the reverse direction. That is minus alpha minus beta So this guy is your h and this guy is your k Adding my point Now the old equation was y square is equal to 4 ax Right So what will happen as I told you change your x with capital x plus h So change your x with capital x plus h And change your y with capital y Plus k. So the equation will now become something like this Okay, of course, there's a foray also sitting over here Got the point But it's not a good practice to write something in capital alphabets. So what do we do? We instead use small alphabets for it Okay, so we write a small alphabet Is this fine any questions Is this fine any questions? So yes, absolutely right those who said change x with x minus alpha and y with y minus beta Okay, of course, you can imagine this from your graphical transformation concepts also So the curve is actually moving see actually the curve doesn't move but when I was teaching you the Uh graphical transformations shifting of origin was not known to us I did not introduce shifting of origin purposefully to you. Okay So if the curve goes right and up, you already know that x becomes x minus something y becomes y minus something Correct, but it is actually because of the origin shifting in a reverse direction Got the point. Okay Now what type of questions you will get on this? Normally the question that is asked on this is they will ask you for the new coordinates They will say what is the new coordinates of the vertex? What is the new coordinates of the focus? What is the new equation of the direct x? What is the new equation of the axis? They may also ask you the lattice rectum length. By the way, let me ask you this Does the lattice rectum length change because of shifting? No, it is not going to change Right, but yes coordinates equations. They are going to change because of the shifting process. Okay So, uh, should we do a exercise here? Or finding the new equations and new coordinates here. Okay, so let let's say For the same situation Let us write down the following vertex focus equation of direct x equation of axis And length of the lattice rectum. Let me write it just lr for the short form Okay, so let's do this so vertex you have already seen alpha beta. Okay Now I would request you to first figure it out on your own and then I will discuss it out Everybody try it out Is it done? Anybody ready with the answers? By the way, am I still audible properly or was the previous one better? Is this better or the I mean or are both of sound quality is same Any feedback? It's the same. Okay, then should we discuss it now? Okay See as I already told you the vertex is at alpha beta. That is obvious. Okay, but I will tell you a Interesting way to write down these coordinates and equations without much hassle. Okay, see if you want to know the vertex right See vertex was originally at zero zero, right? Okay, so write it like this capital x So what do you do first here? See this is something which I call as a role play role change method. Okay role change There's a role change happening. See Your new equation is this Okay, now, what do we do is we normally compare it with this Okay, that means your y becomes y minus beta And your x becomes x minus alpha okay a remain same. Okay Now for this parabola close your eyes and remember everything about it vertex was that origin correct? So origin is something which I would request you to write like this Okay And instead of x who's playing the role of capital x you place that over here So capital role capital x role is being played by small x minus alpha So replace your capital x with small x minus alpha similarly Capital y role is being played by y minus beta So put that here and solve for your x So when you solve for your x you automatically get x as alpha and you solve for your y you automatically get y as beta So alpha beta becomes your vertex alpha beta becomes your vertex simple as that Right, so you don't have to imagine the figure while solving the question many people start imagining the figure Oh, sir, it got shifted like this. So my Vertex would have gone there. My focus would have gone there My director's would have shifted this position all of that you do not need to do Okay, just follow this trick of mine You will be able to find all the coordinates all the equations in a hassle freeway. Okay Next focus I want to find out right so for capital y square is equal to 4ax What used to be the focus Focus used to be a comma zero. So right like this capital x equal to a and capital y equal to zero And just see who is playing the role of capital x Who is playing the role of capital y? And just equate it to a and zero respectively. So your x becomes a plus alpha And your y becomes beta. So now just collect it like this and this will be your focus coordinate Okay, so this will be your focus coordinate Is it fine any questions? So on the figure also, let me show you Okay, this part this focus will be a plus alpha comma beta Is it fine any questions? Any questions Okay, now next part is equation of the directrix equation of the directrix Was capital x equal to minus a Isn't it so in our case capital x is Small x minus alpha correct. So your new equation will become x plus a minus alpha equal to zero. So this will become your directrix equation Okay, so this directrix that you Had drawn over here the equation of this directrix would become x plus a minus alpha equal to zero Let me write equal to properly. Is it fine any questions any concerns? All set Okay, fourth one was equation of the axes So again, close your eyes and just imagine this parabola in your mind In this case your axis was your x axis whose equation was y equal to zero. So just write capital y equal to zero So capital y equal to zero means small y minus beta equal to zero. This itself becomes your equation of axes Is it fine any questions? So this line is y equal to beta line This line is y equal to beta line or y minus beta equal to zero one and the same Is it fine any questions? Next length of the latter symptom as you already know length of the latter symptom is not going to get affected because of shifting So it is still going to be for a So if you have any questions any concerns any discomfort Please do get it clarified right now because once I start doing questions It's going to be very difficult to explain the same things over and over again Okay One natural question can arise in your mind that sir will this work for all the standard cases when they are shifted. Yes So even if you shift y square is equal to minus 4x somewhere But of course, please note that shifting must be done in such a way that The new axis is still parallel to the old axis That is what I mean when I do and when I'm shifting I cannot shift and rotate also in that case all of them will go for a toss So let me write it down here So in this form our new axis our new axis is parallel to the old axis Getting the point Any questions any concerns here? So let's start solving few questions Can you scroll down? Yes. Yes. Why not? Scroll down. I mean I cannot go further down. This is the end of the screen actually As you can see I'm trying to push it up. It's not going here only. Okay. I've heard you have got Circular from the school that from third of January you have to attend hybrid classes Okay, great So till what time will the school run during those days? hybrid classes No time in poor. Okay. So in case it runs till three o'clock. Please do let us know Okay, so that we can shift our Timing slightly later on so that you can conveniently reach home and join the classes Okay Maybe we can delay it by half an hour or 45 minutes or so Why not Erwin? Yes. Yes, definitely we we can always extend it to that No issues And once you receive any kind of a timing information Please keep us updated Through that group Okay, so I have a question based on shifted parabola find Find Number one Let's again find those Five things which I listed down earlier Vertex focus Equation of Directrix Equation of axis Length of Later sector. Okay, and I just take a snapshot of this because I don't have to write it down For this hyperbole, sorry for this parabola So let me give you a parabola here y plus one whole square is equal to eight times x minus two And only press and enter once you have responded or once you have found out the answers to all the five Okay, so I'm giving you around two and a half minutes to do this. Okay. Do it slowly Follow the role change method that we have discussed a little while ago That is going to give you hassle-free answers to all these five questions. Okay Erwin Michelle very good One more minute I can give for the people who are trying hard so that you can wrap up your work Okay, very good Satyam Good Yes Shall we discuss it now? All right, so as I already told you take the role change approach So just call y plus one small y plus one as capital y For a will become eight and capital x will be your small x minus two that means I'm asking you to Just change the roles or do a role change over here So capital x will be x minus two for a will be eight. So a will be two Okay, so keep this in your mind Right now close your eyes and imagine this parabola Okay, it's a right word opening standard case. Okay, so close your eyes Everybody close your eyes And just imagine that parabola which is opening right word. Okay and start answering this question Where do you think is the vertex zero zero? But instead of zero zero right Capital x is zero capital y is zero Okay, and just change the capital s as small x Minus two and change your capital y as small y plus one solve for small x solve for small y and there you go two comma minus one becomes your Vertex coordinates. So those who have got two comma minus one absolutely right Okay, next one again close your eyes back Where used to be the focus? Where used to be the focus focus used to be at a comma zero, right? So instead of a comma zero you just have to do an extra step put x as a and y is Okay, and now replace who's playing the role of what so this guy is being Uh capital x only has been played by small x minus two a is two already And this is y plus one solve for x So this becomes four comma minus one. This is your focus Okay, so if you've got four comma minus one absolutely, right? Next equation of the directories Yeah equation of the directories equation of the directories for such cases Used to be capital x equal to minus a Isn't it? So who's playing the role of capital x small x minus two minus a will become minus two So x equal to zero will become your direct fix Got the point any questions Okay, that happens to be your y axis itself Okay, so this is your direct fix equation Any questions now again close your eyes back What used to be the axis of this parabola? You'll say the x axis x axis means capital y equal to zero So in our case it will become y plus one equal to zero you can leave it like this You don't have to write y equal to minus one. This itself is an equation. Okay, so this is your axis equation correct Now length of the latter system, please understand four a By the way, this constant itself is your latter system. So eight Okay, so four a which is nothing but eight units. This is your latter system Is this fine any questions any concerns to let me know Don't worry. We'll take some more practice on this Okay Now I can see most of you have got everything correct. That means your concept is Sound perfect. Okay, should we go to the next one? All right, let's try this one. I want the same five things Now for this parabola Anybody with any success? Okay, we shall very good Suck them very nice I think yours and vishal's answer is exactly matching Okay, hashita I think same is true for hashita as well. Good Arvind nice Okay, last 45 seconds for the people who are trying hard Okay, shall we discuss now everybody's ready? Okay All right, so let's start first of all with which standard case So this is a shifted version of which standard case Let me ask this question like that that will that that will be more connecting to you So it is a shifted version of y square is equal to Minus four a x right the left-handed parabola or the left-hand opening parabola, correct Why because of this minus sign setting? Okay, so this is the case which you are going to compare it So when you compare you realize that the role of capital y is being paid by small y minus three The role of a is being paid by one Before that, let me write down who's playing the role of capital x capital x or is being paid by small x plus three a is one, okay Now Close your eyes everybody close your eyes and just imagine What are the critical points and critical equations for a left-ward opening parabola? Right, so the vertex used to be still zero zero So vertex used to be still zero zero so zero zero means you will put your capital x as zero and capital y as zero right in other words your small x plus three will be zero and Small y minus three will be zero so x is negative three y is three So your vertex becomes minus three comma three this becomes your vertex correct I hope most of you have got this Next again close your eyes Where used to be the focus focus for such a parabola used to be at minus a comma zero, right? So put x as minus a y as zero Now your capital x is small x plus three minus a means minus one capital y is small y minus three Put it to zero your small x becomes negative four Y becomes three So minus four minus three becomes your focus Is it fine any questions? Okay Next equation of the directrix So again close your eyes For a left-ward opening parabola directrix used to be capital x equal to a Okay, in fact it used to be x equal to a but right now we are writing it as capital x equal to it okay So capital x is nothing but small x plus three a is nothing but one So x plus two equal to zero will become your equation of the directrix Yes, or no okay Next axis axis is still x axis. So you'll say y equal to zero or you can say Y minus three equal to zero. That's itself is your equation. So that's is your axis equation Okay, length of the lattice victim as I already told you the constant that you see over here Of course Without the negative sign that is your four a length Okay, so four a in this case would be nothing but four units Okay, well done. I could see many of you have got it perfectly right not a single mistake anywhere. Well done very good Is it fine any questions? Don't worry. I'll give you one more opportunity. So before a break We will take I think one more question on a shifted case But if you want to copy anything down ask anything Please do so because this is one of the basics of your understanding of parabola. Okay. Good All right Let's repeat the same set of questions Find the following Find the following for let me write it just like this Y minus All right Let's discuss it. Everybody is done Most of you Are still doing it. I can give 45 more seconds. Oh, okay. Okay. I should take your time. Okay, satyam Adna very good Okay Should we discuss it? So basically when we are going to compare this parabola, we are going to compare it with Capital x square is equal to four a y Okay, so who's playing the role of what let us discuss it over here Your capital role capital x role is being paid by x plus two Capital y role is being paid by y minus four And your four a is actually one in this case because if nothing is there you can just assume that there's a one over here So that means your a becomes one by four Okay Now close your eyes imagine that there is an upward opening parabola For your upward opening parabola the vertex is still for this case vertex is still zero zero So your x plus two is zero Y minus four is zero correct Okay, I should So x is negative two y is four, which means Minus two comma four is your vertex. So this becomes your vertex Is it fine any questions? Second keep your eyes closed And I imagine where was the focus Where was the focus focus was at zero comma a So write zero for here And a for this zero comma a so x plus two is zero And y minus four is a in this case a is one fourth correct So x becomes negative two And y becomes 17 by four So negative two comma 17 by four becomes your focus Is it fine Is it fine any questions Next again close your eyes And ask yourself where used to be the directrix for an upward opening parabola So directrix for an upward opening parabola was y equal to minus a In other words small y minus four is equal to minus one by four In other words, you can write it something like this Okay, so four y Minus 15 equal to zero or y equal to 15 by four also that is also fine So this used to be your this will be your directrix Access for this parabola is your y axis. So capital x will be zero, which means x plus two is zero That's itself is your axis well done Okay And finally the length of the ladder system The length of the ladder system in this case is four a units which in this case will become one unit Is it fine any questions Any questions any concerns Now I would not repeat the same type of question rather we will take a different version of the same type So, let me just pick up a question here Where should I find yeah, this is my parabola Yeah, let's take this one Find the equation of the parabola whose focus is at four comma minus three and vertex is four comma minus one Now first make a diagram out of it from the diagram a lot of things will become very evident Make a diagram. Okay. Very good See as I told you make a diagram everything will become very evident from there So it's a vertex is at four comma minus one. Okay So four comma minus one if I'm not mistaken it will be somewhere over here Okay, this is four comma minus one Next focus is four comma minus three Okay, so four comma minus three means further down here Okay, so your parabola is actually how is it can somebody you know guess it You'll say sir. It is actually a shifted version of a downward opening parabola. Am I right? Yes, it's a shifted version of a downward opening parabola like this Okay Now the moment, you know, you know, it's a shifted version of a downward opening parabola immediately You will make the equation to be something Let me write it like this capital x square is equal to minus four a y Correct. Yes or no Now looking at the scenario your capital x will be x minus four. Am I right? And capital y will be y minus minus one which is nothing but y plus one correct And a will be this distance remember a is the distance between the vertex and the focus. So this is your a So a in this case will be two units So now all you need to do is Just fill in this capital x is small x minus four whole square Minus four a will be minus eight And capital y will be y plus one. Of course, it is your wish to expand it and write it. I'm leaving it up to you In fact, I will also, you know, expand it and write it. So that answer becomes x square minus eight x Plus eight y Plus 24 equal to zero. Is it fine any questions any concerns? Okay, so before going for a break one more question. I would like to take on this Okay, let's take this question uh This time Same thing vertex focus data system access direct clicks, but this time the equation has been written not in a form which was you know, uh Given by me earlier. So I was very kind enough to tell you. Okay, this is the way From the from the way I wrote down the equation. It was very evident that okay This was the parabola which was shifted But don't expect the examiner to be as kind as you know, I was when I was giving that question He can actually, you know, expand the entire term mix it up like this and then give you a question Okay, so let's take this as well and see Whether we are able to find out the vertex focus letters, rectum access and direct Try this out everybody. Okay. Chintia. Okay. Chintia. No issues See, uh, we are all please understand here. Oh, you want more time? Okay fine I'll I'll give you more time. No issues. See it's more important for you to solve it Rather than me telling you how to solve it Take time. No issues I can wait Okay, ready. Should we discuss it? Or anybody needs more time? So see look at the question first of all, you would realize that it's it has a Second degree power or degree two on x. Okay. So basically if you convert it To the forms that we have already seen In the shifted form It must give you a quadratic in x kind of a thing, right? So normally how do I solve this question? I just separate out, you know, the Variables like this. Okay, and try to complete a square here. So if I complete a square here, I would need I would actually need 16, right? So just add a 16 On both the sides of the equation so that it becomes x plus 4 the whole square Okay, and send the other things to the other side. So x plus 4 the whole square Take this to the other side to become 12. Take this to the other side to become minus 12 y. Okay now Express this equation like this take a minus 12 form and y minus 1 now, please remember everybody When you are writing the shifted form of a parabola the Direct coefficient of x and y should have actually been made one like how you see over here. Okay I'm saying direct indirectly. It will be minus 12 here in this case. Okay But try to make the direct coefficient as one for both x and the y If it requires something to be taken common, please do it. Please take it Okay Now compare this with capital x square is equal to minus 4 a y And ask yourself who is playing the role of what? So capital x role is being played by small x plus 4 capital y role is being played by y minus 1 And your a is nothing but 3 A is nothing but 3 Yes or no Okay, now again close your eyes. Imagine everything that you can for the downward opening parabola So vertex used to be at 0 0 So write it like this. So x plus 4 will be 0 y minus 1 will be 0 So negative 4 and y will be 1 so negative 4 comma 1 will be your word next. Is it fine next Focus focus for this kind of a parabola used to be at 0 comma minus 8 For downward opening parabola 0 comma minus 8 so x plus 4 is 0 y minus 1 is minus 3 So x is minus 4 y is minus 2 Right, so vertex becomes sorry focus becomes minus 4 minus 2 Is it fine any questions? Any questions any concerns? I can see people making mistakes Okay All right next is Directrix Directrix for such cases used to be if you close your eyes and imagine for x square is equal to minus 4 a y Directrix used to be y equal to a So y equal to a means y minus 1 is equal to a a is 3 in this case So y minus 4 equal to 0. This will become your directrix. Good enough Any questions Access equation for the downward opening parabola used to be y axis so capital x equal to 0 which means x plus 4 equal to 0 becomes your axis correct next Later symptom length used to be 4 a units which in this case will be 4 into 3 which is 12 years Is it fine any questions any concerns? Okay So we'll continue with more of such questions. Maybe on the other side of the break Okay, and on the other side of the break I will also talk about the parametric form of the equation of the parabola and we'll do some questions also based on the same Okay, so please copy this down and if you have any questions, please do let me know else we will go for a break So right now 6 11 We can meet at 6 26 Okay, see you on the other side of the break Now the next part of this chapter. We are going to discuss about parametric equations. Okay, so parametric forms of the standard parabola that we have done. In fact, we can write the parametric form for Uh, you know shifted form as well. So we will start our discussion with parametric forms of the standard parabola So let me begin with the very first of that list which was y square is equal to 4 ax Okay Now again as I already told you in the previous class parametric forms can be numerous Right, there's no sacrosanct parametric form that okay. This is only the parametric form Something like this is not there. Okay, it is going to be like numerous, you know parametric forms are going to be possible So fixed parametric form. There's nothing like a fixed parametric form You can write parametric forms, you know numerous parametric forms for the same curve But normally there is one parametric form which everybody uses Okay, and that actually becomes a convention also Okay Uh, if you would recall in your statistics if you recall your class 10 statistics While we were basically taking that, you know Assumed mean method to solve the mean of a data Our teacher used to always take the assumed mean as one of the class marks, right But is that Necessary no it is not necessary. It is just a convention that people follow because it has got some add-on advantages Okay, in the same way parametric form Is not a sacrosanct value. It's like you can have numerous kind of identifications Your a hard card number can act like an identification your fan card number can act like an identification your first certificate So basically parametric form is nothing but it is trying to give an identification to every x comma y Point lying on that. Okay. So one of the parametric form which is quite recommended for this kind of a parabola is This please note it down Okay, so this is a suggested or recommended Let me write it on the right side. Yeah, this is a suggested or recommended parametric form Where t is a parameter Okay, and if you try to eliminate your t, you will automatically get y square is equal to 4x Okay, is it fine? So if you try to eliminate your t, you will automatically get y square is equal to 4x So as you already know the use of parametric form is in Is in denoting a point on that curve So you can represent a point on that curve with minimum number of variables involved Okay, for example, let's say if I want to choose a point on this curve y square is equal to 4x I would better choose this point or rather choose this point as at square comma 280 Okay, as you can see only one unknown t is involved over here Right, if you choose a generic point x1 y1 unnecessarily two unknowns will be involved and that will make your problem solving slightly more complicated okay All right, so uh, I would now request you all to suggest me Okay, take a clue from this and somebody please suggest me The parametric form for y square is equal to minus 4x What is the parametric form for this? Anybody see yes, as I already told you while I was discussing the standard form that Uh, this equation the second one is obtained from the first one by changing your sign of x Isn't it your x has been replaced with a negative x In the same way in parametric form. Also, you do that change your at square As minus at square. That's it others Won't change Okay So this becomes your parametric form for this where t is a parameter By the way parameter you can write anything I have used t You can write t you can use any way any you know name of the alphabet it is up to you Okay all right Let's now Let's now suggest a parametric form for x square is equal to 4 a y anybody x square is equal to 4 a y Right, just start the position of x and y so x will become 280 and y will become 80 square Okay, so this will become a parametric form. Of course t again here is a parameter T again here is a parameter Is it fine any questions fourth one x square is equal to minus 4 a y Now you'll see it's that simple change the sign of y. That's it Yes, so x is equal to 280 And y is equal to minus 80 square. Okay Where t is a parameter Is it right? So just keep these parametric forms into your mind because once we start doing these concepts form your competitive level exam point of view We'll be using these parametric forms in order to solve many questions because the the main use of parametric form is It facilitates us to choose a point in a hassle-free manner On the curve. Okay, that is the main purpose of a parametric form Okay all right So based on this let's take a question So I hope everybody has noted down the parametric form for all the four standard forms Now a small question Suggest a parametric form for Suggest a parametric form for x minus 1 whole square is minus 8 y plus 1 Okay, suggest a parametric form for this Again, I have used the word suggest because there could be several parametric form that you may think of Okay, so please tell one of the forms that you feel Very good. So you can take a clue from here. You'll your a will become 2 in this case so you can use this form itself So you can say x minus 1 is 280 280 will become 40 That means x is equal to 40 plus 1 And y will become 8 minus 80 square which is minus 2 t square That means y can be written as minus 2 t square minus 1 Isn't it? So this finally becomes your parametric form for this curve where t is a parameter Where t is a parameter. Is it fine any questions? Shall we go to the next question? Okay, let me give a question for my side here. Okay, so let's say this is our Y square is equal to 4 ax parabola. Okay Let's say this is your focus s and this is a focal chord So pq is a focal chord Okay point p The parameter is t1 Okay, that means point p. Oh, I'm so sorry point p. The coordinates are 81 square comma 281 In other words, you're trying to say that point p has a parameter t1 And point q is let's say a t2 square comma 282. That means it has a parameter of t2. Okay Prove that if pq is the extremities of the focal chord prove that if pq is a focal chord then The product of their parameters Of these two points p and q should be multiplying to give you one. Please do this everybody And this is one of the properties which you are going to use very very commonly while solving many questions. So keep this in mind Also later on when we are doing this concept from your competitive level point of view We'll be using this result very very frequently So again, I'll repeat the question pq is a focal chord T1 is the parameter for the point p. Please remember Every point has a parameter for it. So parameter is like the adharkar number for that point. Okay So the adharkar number or the parameter for p is t1 And the parameter for q is t2 Okay So prove that t1 t2 is going to be minus one Very simple. It is not going to take you much time Use the fact that p sq is collinear. That's it. That's all you need to do and let me know with that done if you're done Okay, done sattva. Very good. Anybody else? anybody else See very simple. You have to use the fact that points p s and q are collinear correct That is to say that slope of sp Should be same as slope of sq Okay, so what is the slope of sp? y2 minus y1 by x2 minus x1 Can I say the same will be applicable for the other slope as well, but it's just that your t2 will come in place of t1 Oh correct yes or no Now let us try to simplify this equation over here You may first cancel out 2a and 2a and a and a form your numerator and denominator respectively Something like this Let's cross multiply Okay, let's open the brackets. You get t1 t2 square minus t1 Is equal to t1 square t2 minus t2 correct now bring This term to this side and this term to the other side So you'll end up getting t1 t2 square minus t1 square t2 is equal to t1 minus t2 Now here take t1 t2 common So if you take t1 t2 common, this is what you're going to get by the way right side also you can write it as this Now cancel these two off So we'll end up getting t1 t2 as a minus one Okay, now many people asked me said t1 could have been equal to t2 But please note t1 t2 are not the same because they are parameters of Two distinct points p and q Okay, as I hope you can all see the figure T1 and t2 cannot be the same as I told you two points cannot have the same parameters It's like two persons having the same adharka number Right, it's not possible. Isn't it so here t1 and t2 cannot be equal So they can be cancelled off. So once you're cancelled off you are left with t1 t2 as minus one Please keep this in your mind and remember this result. Okay now Once we have achieved this, let us go back and analyze something very interesting But before that if you want to copy down anything ask anything, please do so Okay, all right So basically if you again draw the figure, let me draw the figure once again So basically what we realized from this small exercise is that If there is a focal chord Okay, if there's a focal chord Okay, where one on the one end of the focal chord is a t1 square comma 2 a t1 Then the other end of the focal chord will be nothing but a by t1 square comma minus 2 a by t1 Right because t1 t2 is minus one So t1 t2 minus one gives us t2 as Negative reciprocal of t1 correct So ideally if this point would have been a t2 square comma 2 a t2 and you would replace your t2 with negative one by t1 then this point P q coordinates would have actually become this Okay So in other words the parameters and the extremities of a focal chord are negative reciprocal of each other For example, let's say if this point was let's say I took t1 as a 4. Sorry t1 as a 2 So this this point was 4 a comma 4 a Then this point would have been now remember a by 2 square which is 4 Minus 2 a by 2 Okay, so this point would have been a by 4 comma minus a getting my point Okay, so if I know the parameter at one end of the focal chord By the virtue of the fact that t1 t2 is minus one I would automatically know the parameter at the other end as well. Is it fine any questions? Any questions? Can I go to the next slide? Now the next question that I would like, you know us to solve in fact, you know I will be showing you something very interesting I will be proving now that lattice rectum is the shortest focal chord I will now be proving Here that lattice rectum is the shortest focal chord So all of you please pay attention The proof is very simple. Just have to be attentive to it Let's say this is our standard form y square is equal to 4x And let us say this is our Any normal focal chord. Okay. Now we have already seen that If this point is a Let's say I call it as a t square comma 280 Then this point will become a by t square comma minus 2a by t. Correct? Yes or no Correct. No, everybody's convinced with this Now What is sp length? Now sp length, let us say I assume that this is my directrix Okay, we already know that sp length is same as pm length Correct because it's a parabola In a parabola The definition of the locus definition of the parabola says sp is equal to pm. That means the distance from the Focus should be same as the distance from the directrix. Okay. Now everybody, please pay attention This is at square and this is a So how much is your pm length? You will say a plus at square. Am I right? Yes or no Anybody has any doubt with respect to the length of st Any questions anybody with the length of sp No, okay. So write down on the chat box. What is the length of sq on the chat box? Write down length of sq I've actually drawn it also for your reference so that you can take a hint from there anybody are you Same as qm Right now this length is a by t square. This length is a This is a by t square. So what is the length? a plus a by t square. No Yes or no Hello, I'm audible everybody. Nobody's responding Okay. All right. So what is this length of the focal cord focal length is sp plus sq Which is nothing but a plus at square And a plus a by t square Correct Which is nothing but to a plus Yes or no Right now everybody please pay attention very interesting Do you all recall that for positive numbers am is greater than equal to gm? We have done this in our sequence series topic Now focus on the two numbers being at square and a by t square. So let's say we apply On this you apply am greater than equal to gm. Okay So we are all convinced both are positive numbers because a is anyways positive and you have squared up t So can I say this term? by two Will be greater than equal to the geometric mean correct In other words at square This by two will be greater than equal to a square and the root is a that means This fellow is greater than equal to to a Right, everybody's convinced here Everybody's happy with this Yes now, okay. So now use this over here Apply it at this stage. So can I say this term will be greater than equal to two a plus this is two a In other words pq will be greater than equal to four a Now whose length was four a if you recall Later symptom length was four a That means any focal chord that you make its length will be greater than equal to four a Which means the least value of pq will be four a yes or no. What does this convey? This conveys that lattice rectum is the shortest focal chord Yes, or no. Yes or no. Please tell me Is it convincing enough any questions any concerns here? So lattice rectum happens to be the shortest focal chord you cannot get any focal chord shorter than the lattice Okay, so if you get a chord, which is lesser than this Later symptom it is not a focal chord. It is a chord. It is not a focal chord So lattice rectum is the shortest focal chord. I'm not saying just chord focal chord. Is that fine? Okay Any questions anything that you would like to copy ask Please do so If not, we'll go to the next question. Let's take another question Read the question carefully if required read it twice The question says Proof that the semi lattice rectum of the parabola Y square is equal to four ax Is the harmonic mean between the segments of any focal chord of the parabola Okay, now let me explain this to you. What does it mean actually anybody who has understood the The meaning of the question I mean, let me just Ask you first of all has anybody understood the meaning of the question No, right. Absolutely. So this question says, okay. Let's say this is your Focal chord, right So let's say p Q s, okay. So what does the question say? The question says sp Two way And sq so you need to prove that these three are in harmonic progression Prove that the semi lattice rectum length is the harmonic mean. So this is the mean of The segments of any focal chord. So these are the two segments sp and sq Okay, that is to say, let's say this is l1 This is l2 then l1 Two way and l2 are in hp. Can you prove it? Can anybody prove this? I'm giving you some time once done. You can say I'm done on the chat box Yes, anybody Done. Satyam is done anybody else See, guys, I don't know why this taking so much time. I've already discussed that Okay, let's call this as Let's call p point as a t1 square comma Or let's call it as a t square on the yt1. Let's call t1 by two ways One more alphabet there And this is a by t square comma minus two a by t correct now Now we have already seen in the previous You know result where we were proving that the lattice rectum is the shortest vocal chord that this is a plus at square and l2 is a plus a by t square Okay Now if you want uh to prove that these three are in hp that means you need to prove these three are in ap correct Okay, which means that you have to prove that One by l1 plus one by l2 is two by two a in other words. You have to prove that one by l1 plus one by l2 is one by Correct, let's check So let us write one by l1 first So that will be this Okay, let's write one by l2 also So this will become t square a plus a t square correct Let's add So if you add The denominators are same. So numerator will get added up So take a common from the denominator and cancel it out What is left one by a done So one by l one by a And one by l2 are in are in Ap one by two Okay Is it fine any questions? So these three are in hp Okay, so for the next 20 25 minutes of our class we'll take some more generic problems just to you know wind up this concept See, uh, we have just covered up uh, we have just covered up As I told you 20 percent of the concept there are many more concepts to be covered in the parabola As I told you in the beginning of the session, we are just doing it for our school exams purpose In fact, we have gone beyond the school as well. So after knowing so much You should not have any issue with the school questions, but this is not the end of the chapter You're just you have just done 20 percent of the topic So tip of the iceberg you have done 80 percent is still left Okay All right, let's take this question. I'll find the equation of the parabola with the lattice rectum joining these two points The lattice rectum joining these two points If you know once you're done with this You may just say are done or you can give me a response also on the chat box both are fine with me Uh, sattam. Do you think only one answer is possible? Think hard and how come there is square on both the sides? Okay, by the way Okay Since I've you know spoken about it. Let me tell you there could be a parabola like this as well. So white and yellow Sorry, it's green color. I know yellow color. Yeah, there there could be two types of parabolas Okay All right done. So let us discuss it See first of all, you know that your this is your focus point. So focus point is 3 comma 2 Okay, and let me also tell you that this length is 4a Okay, so as per your question This length is 8 so 8 is 4a so a is 2 correct Now remember there are two possibilities One is you could have your vertex at this position v1 position Or you could have your vertex at v2 position Okay, if you consider your vertex to be at v1 position You have to go two back from here two back from here means you will end up getting at 1 comma 2 Am I right any questions any concerns? Okay So this is a case where your parabola was y square is equal to 4ax And this was shifted To 1 comma 2 so your y will get replaced with y will get replaced with y minus 2 x will get replaced with x minus 1 So your answer will be for the yellow one for this one your answer would be Is it fine any questions any concerns, but this is only one of the answers Is it fine any questions any concerns anybody has? Okay, if you consider your vertex to be at v2 you have to move two units in this direction So that will become 5 comma 2 So in that case, please note that that would be that would be a case of leftward opening parabola. So something like this Where your y would be replaced with y minus 2 And x would be replaced with x minus 5 So this is your second case. So both of these are your answers to the question This and this Is it fine any questions Any questions anybody? Okay, let's take the next one Find the Equation of the parabola whose axis is parallel to the y axis And passes through these three points And also determine its lattice vector Length of the lattice vector Let's do this question Yes, any success anybody See all of you please pay attention here Read the question first of all the question says you want a parabola Whose axis is parallel to the y axis? Okay, that means your parabola could be of this nature something like this Or something like this Correct Now, isn't it like saying you are writing a quadratic equation here? Because quadratic equation also follows the same nature, right? Isn't it So this is something which you can connect it to your quadratic equations Your quadratic equations are nothing but parabolas whose axes are parallel to the y axis Correct So you can say that this is a representative of a parabola So it represents all the parabola whose axis is Parallel to the x axis Sorry, by axis Okay In the same way, I would also like you to note this down that Such equations will represent parabola whose axis is Whose axis is Parallel to the x axis Okay Now Oh very good Vishal very good. Now in order to know a b and c you have been given three points Let's use that. Let's use that three points to get a b and c So first of all, let's use 0 comma 4 0 comma 4 means c is equal to 4 Am I right correct? So that constant term is 4 Okay, next 1 comma 9. So this is 9 Okay, and then 5 comma 4 5 comma 4 will give me something like this Let me write it in white only Okay, now since your c is 4 you can say a plus b is 5 Okay, and 16 a plus 4 b is equal to 1 Okay, let's multiply this with the 4 So 4 a plus 4 b is 20 And 16 a plus 4 b is 1 Okay, subtract the results Okay, subtract the results you'll end up getting minus 12 a is 19. So a is minus 19 by 12 Okay Put it in the first one. So a is equal to or sorry b is equal to 5 minus a b is equal to 5 minus a And a is minus 19 by 12. So am I right when I say it is going to be 79 by 12 b value Correct So your final answer would be y is equal to minus 9 by 12 x square plus 79 by 12 x Plus 4 equal to 0 in other words 12 y Minus 19 x square plus 79 x plus 48 is your required parabola Well, then Vishal Vishal got it absolutely right very good Is it find any questions Any questions any concerns related to the solution of this problem? Clear everybody. Let's take the next one. Yeah, so this question says If the vertex and focus of a parabola Are at these two points then find the equation of the parabola Then find the equation of this parabola This should be simple because we have already done a similar question before as well Let me put the poll on also so that if you are done you can put your response on the poll as well Yeah Okay, sathya. We'll check Just five of you have responded so far almost two minutes going to be over Okay, let's give you one more minute. Let's have one more minute to solve this Okay, should we discuss it now five four three two one Go, okay. Just eight of you have given the response. Let's discuss it out Out of the eight that have voted five of you say option c. Okay, let's check See making a diagram really helps in this case So vertex is at three comma three. So this is your vertex position Okay And focus is at minus three comma three. So somewhere over here So your parabola is somewhat oriented like this Okay So it's a clear cut case of a leftward opening parabola which has been shifted Isn't it? So a leftward opening parabola helped me shift it in this case Correct So now for that we need to know what is the a value first of all So a value here will be nothing but please note that this distance is your a value It just sticks in this case So y square is equal to minus four a x And in this you have to replace your capital y with y. I'm so sorry. This is three Yeah, y minus three and change your capital x with x minus three So when you do that it becomes y minus three the whole square minus 24 x minus three That's it expand it Let's see. What do we get out of this? Okay, so on simplification this gives you y square minus 6 y Plus 24 x minus 63 equal to 0. Is there any option which matches this? y square minus 6 y plus 24 x minus 63 minus 63 will match with option number c Okay Absolutely right option number c is the right option in this case Is this fine any questions any concerns here anybody? Okay, let's take one last question for the day And we'll be done with the parabola chapter as well Okay, very specific question here. What is the equation of? The directrix of this parabola Very specific question Okay, and it's a shifted form of a parabola only we can all sense it from the equation So let's do this and then we will wrap up the day Uh poll is on poll is on Okay, saptam. We'll check Uh, can we wrap this up in another 30 seconds? I can see only six of you have responded to this so far 30 seconds more Please try to finish it up in 30 seconds. Okay, five three two one Go Oh my god That's a very very mixed response which you have all given me two two three That means there is an element of confusion related to shifted cases of parabola. See The presence of an x square term clearly shows that there is a perfect square which is supposed to be made in x Okay, so Let's make a perfect square. So for that, I need a four here. So add a four here add a four here So this will become x minus two the whole square Minus eight y and uh, this will actually become an eight equal to zero So take it to the other side Right now compare this with capital x square is equal to four a y. So a roll is being played by two Capital x hole is being played by x minus two And capital y roll is being played by y minus one, okay Now for such cases For such cases your directrix was supposed to be this is what they're asking us, right? Okay, the question asks us for the directrix, right? So directrix for this case used to be your y equal to minus a Isn't it because it's an upward opening parabola So y equal to minus a is your directrix So y is small y minus one minus a is minus two In other words y equal to minus one is your answer to this question I think that matches with option number c. Okay, so most of you went for c but most of you made mistakes also, right? So with this we close this topic as well from your school level point of view We'll catch up this topic once again post your Uh exams are over. Okay Thank you. Bye. Bye Take care. Stay safe. Good night to all of you