 So the next sense that we are going to talk about is hyperbola Okay Again, we'll restrict our self only to your school and slightly more than the school Okay, so first of all, what is the locus definition very similar to what we had for an ellipse So hyperbola is basically nothing but it's a locus of a point or a path traced by a point Traced by a point moving in a plane moving in a plane in such a way In such a way that The ratio of its distance From a fixed point to that form of fixed line To that form a fixed line is a constant Where this constant value is more than one Okay, so just like what we had in ellipse It's a ratio again of what distance from a fixed point which we know is the focus To that form of fixed line, which we know is the direct mix Okay, and this ratio is what we call as This constant is what we call as the eccentricity and in this case your eccentricity is more than one In this case your eccentricity is more than one Okay So let me draw a quick Representation of the locus over here. So let's say this is your fixed point s and this is your fixed line Let's call it as D equal to zero on this point P is moving in such a way that This distance divided by this distance, so that's if this is M So SP by PM is a value which is more than one which is more than one Okay, so the path traced by this point P will be that of a structure like this Okay, which will be called as a One of the arms of the hyperbola Okay, remember Maintaining the same ratio there could be one more set of point and a line Such that the same ratio E which is more than one Right, but it could be any number more than one. Okay is maintained That means in case of a hyperbola In case of a hyperbola just like we had it in the case of an ellipse There could be two pairs Sorry, they could be one pair or they could be two sets, right word to use is two sets So let's say s1 and D1 equal to zero and s2 and D2 equal to zero Okay, so if I take a point over here Then this distance s1 P Divided by this distance that is PM1 will be the same as s2 P divided by PM2 divided by PM2 So s1 P by PM1 would be same as s2 P by PM2 and this ratio will be E only The same E that you have chosen for that particular hyperbola There's a lot of difference. I mean structural wise it may look to be the same in a hyperbola that ratio is equal Okay, so in sorry in a parabola this ratio Prakul this distance, let's say I call this point P This distance is equal to this distance Okay, so this point P is moving in such a way that it is always maintaining the same distance Okay, and in no case it can go beyond the Line right as I told you this thing come on this side Right because if it comes the distance from the line will be shorter as compared to distance from the point Okay, so there's a lot of difference and I think I should actually plot it for you to realize that difference I'll just show you How different they look so let me just take a case Let's say we have y square is equal to let's say Four times X minus two Okay, and now I will take a hyperbola x square by by four minus y square y square by My square by one equal to one Okay I purposely chosen it in such a way that you can actually realize the difference of the two arms Okay, your hyperbola is more sharper in a feature Okay, it is basically they are the cases where if you extend the eccentricity to a higher value They will start coming closer and closer till they become a pair of straight lines Whereas in the parabola the nature is quite different Okay, the nature is quite different. So if you see this is actually showing you how the arms can look different But again, it depends upon your A and B value chosen also Okay, so they may look very similar, but they're actually not There's a quite of a difference between a parabola structure and one of the arms of a hyperbola fine, so your hyperbola Can have two arms one looking like this and other is looking like this They both extend indefinitely right, so the purpose of referring To it. I will just make a diagram and keep it. By the way, the terms that we are going to use is exactly the same as what we use for the case of ellipse These are called okay, let me take it in the next slide terminology used for a hyperbola So what I'm going to do I'm just going to borrow the figure from this because I've already drawn it here Let me just shorten it up Let me use this Okay Now as you all know that Let's say this will be the focus and just assuming a certain point to be focused by the way I could have also drawn the focus here just a second focus for the second conic EQ 2 Let's take this as a So these two are the focus or to foresight Okay, so these are called the foresight s1 s2. Let's call it Okay, the directories will be somewhere. I don't know. I should have drawn the directories also Let's check directories of equation This could have been a better figure All right, so these are the two foresight As you can see a and b. These are the two foresight s1 s2 These are your two directories d1 equal to 0 d2 equal to 0 Okay, please note that The midpoint of the two foresight is called the center of the hyperbola. So this will be the center of the hyperbola The line which is connecting the two foresight and also passing through the center of the hyperbola that line is basically called as the That line is basically called as the transverse axis transverse axis Okay, there is no major minor in case of a Hyperbola. Okay, please do not use the terms major and minor for a hyperbola Major minor is used in case of an ellipse not for hyperbola. Okay The line which is perpendicular to the transverse axis and passing through the center That line is called as the conjugate axis Why does called conjugate axis will learn in some time? Okay And other things will definitely remain the same for example if I connect any any two points Okay, this will be called as a chord Remember your chord can also be formed when you connect a point from one branch to other branch. This will also be a chord Okay, so both are chords only So this is a chord. This is a chord chord need not be Connecting two points on the same branch. Okay, so it could connect like this also Let me tell you these are not two hyperbola These are only one hyperbola two branches of the hyperbola. Fine. Do not get confused If a chord is passing through the focus Okay, that will be called as a focal chord. So there'll be two focal chord Sorry, they will be multiple focal chords, but you can draw the focal chord here also Okay, so focal Okay, other things like Double ordinate so any line which is perpendicular to the transverse axis and perpendicular will be called as a double ordinate double ordinate If a double ordinate happens to pass through the focus, I'll just draw a small one over here So Yeah, so these are called your lateral recta. These are called your Lateral recta, is it fine now few things that you will know notice here as Different from ellipse is in the case of in the case of ellipse your vertices where farther away From the foreside that is for say was closer to the center as compared to the vertex You know vertex is closer to the center as compared to the foreside Okay, second thing is again your directories were away from the center as compared to the foreside Here directories will be closer to the center as compared to the foreside All these subtle changes happen because your E is more than one. We'll see how it happens We'll try to analyze it through our coordinate geometry fundamentals So please note down these are the terms that we normally use Nothing much of a change here other than the fact that you call the axis connecting the foreside as transverse axis and The axis perpendicular to the transverse axis and passing through the center as the conjugate axis There is no other change other than use of this two particular Name of the axis. Okay Now let's try to talk about the coordinate geometry fundamentals So we are trying to find out the equation of the standard form of a hyperbola So let me just draw a quick diagram so that I can use it now This is your transverse axis and this is your conjugate axis in order to Basically find out the equation of a standard form We will assume that our x-axis is oriented along the transverse axis and our y-axis is oriented along the conjugate axis Okay, so as a result what happens the center becomes the origin Okay So just like we did it for the case of an ellipse will assume that the vertices are at a comma zero and minus a comma zero Okay Focus for the time being we'll take it as c comma zero and minus c comma zero Okay, and these points I will take it as minus d comma zero and d comma zero sorry other way around D comma zero and minus d comma zero Let me name them also so that I can refer to it. This is s1. This is v1. This is n1. This is center This is n2. This is v2. This is s2 Okay Now as Discuss in the ellipse. We don't want extra terms like c and d So I will try to write them in terms of a and e just like the way I did for the ellipse Now for that I will choose this point v1 as my point lying on the hyperbola So as for the definition of a hyperbola, we know that hyperbola is such a point where the distance of a Point from the foreside divided by distance of the same point from the directrix Okay, this is of the same point from the directrix is equal to the eccentricity Correct, and this e is actually more than one or a hyperbola Similarly, if I take the point distance from s2 divided by the points distance From n2. This should also be so very same point distance from s2 Divided by the same point distance from n2 or this directrix should also be okay So the first one gives me The first one gives me c minus c minus a By c minus a by a minus c by sorry Yeah, c minus a and this distance will be a minus d a minus d my bad is equal to e Remember this distance is a minus c Okay a minus Sorry c minus this is c Minus This distance which is a okay Divided by distance of the same point from the directrix Which is a minus d so from here I get c minus a is equal to a e minus d From the second equation similarly, can I say c plus a by a plus d is equal to e Okay, remember the distance of Remember the distance of v1 from cs2 will be nothing but a Plus c1 plus c. Okay a plus e so which is numerator term and distance of v1 from n2 will be a plus d Okay, so from here. I will get c plus a is equal to a e plus de So from these two equations, let me call them as one into If you add them You will get to see is equal to two a e so c becomes a e and if you subtract them That means if you do 2 minus 1 you will get to a is equal to 2 de that means d is equal to a by e So going forward, we will not call it as c We will actually call it as a e and you'll see that the same term was also coming for an ellipse Okay, and going forward. I will not call it as a d. I will call it as a a by e Okay, as a result the director says now will have the equation x equal to a by e and x equal to minus a by e Now the reason why director says are closer to the center is because you're dividing by e which happens to be a quantity greater than 1 So the whole quantity will become lesser than a as a result They they are closer to the center as compared to the vertices In fact, they are closer to the center as compared to the foci because foci you are multiplying a with e Are you getting my point? And the opposite was happening in case of a ellipse Okay, then if you divide by as quantity which is less than one that will become farther away from the center That's why your director says where farther away from the center as compared to the foci and or as compared to the vertices Getting my point so now having got these critical points and having got the equation of the Directises, let us find out its equation So I'll go to the next page I'll pull this diagram again Okay, so let's say this was our One of the foci Okay, and this is one is this is the directix corresponding to it. Let's take a point h comma k So what is the locus definition of a hyperbola the locus definition of a hyperbola is the ratio of sp to p m is E that means sp is e p m What is sp sp is under root of h minus a the whole square plus k square is Equal to e times p m now. What is p m over here? P m is nothing, but it is going to be h minus a by e mod correct H minus a by e mod Now you may introduce this e inside like this and Let us square both the sides Let us square both the sides So that gives us x h minus a the whole square plus k square is equal to each minus a the whole square That's nothing, but it's square plus a square e square minus two h a e Similarly here e square h square minus two h a e plus a square these two terms will get cancelled off These two terms will get cancelled Okay Now let's bring this a square to the other side and let's bring this h square to the other side and let's bring Cray square also to the other side So you will have e square h square minus one minus k square as a Square e square minus one. Let me know if you have any problems to the stage Now divide by a square e square minus one. So let's divide by a square e square minus one So when we do that We'll end up getting oh, sorry. It's the other way around. Yeah Yeah, it's h square e square minus one. Thank you, Ritu. Thank you So when you divide by that you get h square by a square minus k square by a square e square minus one equal to one Okay, let's generalize this Let's generalize this by replacing our h with an x and k with a y So we'll end up getting x square by a square minus y square by a square e square minus one equal to one Okay, now for the purpose of convenience, we will write this guy as a b square Now again, why b square? How are we sure that it's going to be a perfect square? Remember, what did I discuss with you? He was greater than one Okay, so e square will also be greater than one. So e square minus one will be positive So a square e square minus one will also be positive and hence there is no harm in calling this as a b square Right. So here you have a equation in front of you Which is one of the standard forms of the equation of a hyperbola. Now again, please Note that there is nothing like a greater than b or b greater than a In the same situation, you can have a greater than b also You can have b greater than a situation also That doesn't change the orientation of the hyperbola hyperbola will still be in this fashion where your Transverse axis will be along the x-axis and your conjugate will be along the y-axis Even though a is greater than b or b is greater than b Are you getting my point? Let me quickly show you right now. You can see on your screen. I have drawn a hyperbola where I have shown a is more than b even if I draw Let's say even if I draw Be greater than a Orientation is not going to change. Yes, their concavity is going getting changed as you can see this this para this hyperbola is No, slightly more bigger, right? Mean the arms are slightly more open But orientation is not changing Orientation in the sense that your transverse axis is still the x-axis Your conjugate axis is still the y-axis, right? So a greater than b greater than a doesn't make it, you know other way around We'll talk about the other way around also that is something else, which is not the case here, okay? Is this fine any questions? Any questions with respect to this? Okay, so few changes that you would see from the equation of an ellipse in the ellipse There was a plus sign over it actually there's no difference from the ellipse equation Because in an ellipse you had one minus e square coming up over it and there was a plus sign So there's absolutely no difference other than the fact that how we represent it is slightly different Here we take b square as a square Times e square minus one in the other case of the ellipse We used to take b square as a square one minus e square That's the subtle difference that you will see and that's why many a times you will realize when we do it in more detail That most of the ellipse nature is being taken by a hyperbola where you just replace your b square with a minus b square That's something which we'll talk about when we do the chapter in more detail as of now Please make a note of this Here you have to be careful Pakul the chances of you making a mistake is very very high. So Pakul was asking about Ascenticity when you are finding for this case remember your I'll just come from the same expression again. So your b square is a square e square minus one So e square minus one is b square by a square. So e square is one plus b square by a square And e becomes under root of one plus b square by a square Now here There is a lot of chance that if you do one plus a square by b square instead of b square by a square You'll still get an answer which is more than one, but it will not be your Ascenticity of the hyperbola. So the chances of you making a mistake here is You know there in case of an ellipse there was no chance at all, right? But here you can make a mistake. So here you have to be careful. That's what I can see Okay now coming to the equation of sorry the coming to the Later spectrum length, okay, let's talk about latter-sector length here itself. So I'll just take a latter-sector here Okay, so latter-sector is basically a double ordinate which passes through the foresight So let me take L1 and L2 as the extremities of the latter-sector. So it'll be minus a comma something Okay, and a comma something Sorry minus a comma something. Let's take all this as k. Okay. Let's figure out. What is k? So I will use the very same equation. I'll use the very same equation So a minus a e square by a square Minus y square by b square equal to 1 so this will simplify to This will simplify to Y square by b square is equal to e square minus 1 now remember e square minus 1 is b square by a square So y square is b to the power 4 by a square. That means y is plus minus b square by a So again, you realize the result is the same as what we got for an ellipse So that's why we say if you know your ellipse formulas properly even your hyperbola will be taken care of But with small minus changes. Yes, okay So the length of the latter-sector here would be the distance between your L1 and L2 Which is to be square by a units Is it fine any question with respect to the length of the latter-sector? Is it fine can I move on Okay, so this is one of the standard forms and just to recap in this standard form Okay, in this standard form the equation is x square by a square. I'll just write it on the other page So in this standard form okay Your transverse axis is along the x-axis conjugate axis is along the y-axis So orientation of the hyperbola will be like this Okay Remember your foci will be at a comma zero minus a comma zero Okay, center is at origin. By the way, let's say I call this as the vertices v1 and v2 this length v1 v2 is called the Length of the transverse axis Okay, length of the transverse axis here is to a units Okay, now is there anything called the length of the conjugate axis But there is no intersection of there's no intersection of the hyperbola with the conjugate axis So how can we find the length over here? Now all of you plus pay attention Why it is called conjugate axis. Let us try to address that concern See if you take this hyperbola Okay Now you know your a and b are real quantities Right because a and b are related to each other by the formula b square is equal to a square e square minus 1 right? So e is real one is real a square is real. So b is also real Now here if you try to find out where does this hyperbola cut the y-axis that means try putting x as 0 If you try putting x as 0 you will get something very it's strange You get something very strange you get y square is minus b square Thereby suggesting that square of something is actually a negative quantity Now b is definitely positive. So y has to be imaginary in this case, which is plus minus b i Right so actually your y-axis is being cut by the hyperbola at imaginary points And those points are actually b i and minus b i Since b i and minus b i in the language of complex number are known to be conjugates That is why this axis is given the name as conjugate axis, but it is not a real cutting Because you cannot see it Right when you're drawing it on a when you're drawing it on a real Cartesian coordinates You cannot see this cutting happening because if you're seeing that cutting happening That means those two points are having real coordinates not imaginary coordinates in this case They are cutting which you cannot see and since you cannot see we say they're cutting at imaginary points So which are those points b i and minus b i? Getting my point okay, and that is why the name is given as a conjugate axis to it because here the Hyperbola cuts the axis in two points which are conjugates of each other Now for the purpose of just mentioning we say that this point. Let's say b1 is b2 are these two points So b1 b2 length, which is actually to be length is The length of the conjugate axis, but this is hypothetical Okay, this is hypothetical because you cannot see this length you cannot see this cutting happening Okay, so hypothetically we say that the length of the conjugate axis is to be units Is this fine make sense here only some people feel a slightly you know disconnect from it But here is the instance where you are seeing a real coordinate system, and you're also talking about a Imaginary axis at the same time Okay, so imaginary axis is being used over here fine All right again I would like to tell you there is nothing like a is more than b or b is more than anything can be more than the other It doesn't depend on that. It doesn't depend on that Okay So with this I will introduce you to one more standard form So this is one of the standard forms just like in hype ellipse We had two standard forms in hyperbola also there will be two standard forms one of them is what you are seeing right now on your screen Okay, the other standard form is the one where the Transverse axis is along the y-axis and Conjugate axis is along the x-axis that means your hyperbola appears to be like this Okay, please draw arrows over here. So this becomes your transverse axis and this becomes your conjugate axis Okay, but what we'll do is we'll count will continue with this Nominclature will call this point as 0b will call this point as 0 minus b So your vertices are 0b and 0 minus b Your foci I will take it as 0 be and 0 minus b Okay, and let me show you the directresses also. Let's take this as your directresses So directresses is y is equal to b by e and y is equal to minus b by e Fine Now for such a case if you start deriving the equation you realize that I'm just saving I'm just saving your time over here Your equation will look like this x square by a square Minus y square by b square equal to minus one will come Many books will write it as y square by b square minus x square by a square equal to one But what I follow and I will recommend you to follow you just follow the same left-hand side version as what you followed for the previous But one will now become a minus one One will now become a minus one and here your b square. Sorry here your a square will be B square times e square minus one so small changes that you are going to notice over here So now your a square will be b square times e square minus one in other words e square minus one will become a square by b square So e square will become one plus a square by b square. That means your e will become under root of Okay, so this is what prakul was asking. So, how do we refrain ourselves from choosing a wrong position of a and b? You have to be careful all I can say So whenever you realize that in your equation, there is a minus one appearing when you are writing it like this It's a case of the actually by the way, this is called a conjugate hyperbola We normally call this as the conjugate hyperbola. Okay. In fact This hyperbola and the other hyperbola are conjugates of each other. That is what we should call it But yeah, most of the things are basically same as in ellipse It's just that your b square is getting changed with a minus b square prakul, right? So all your formula that you have seen for an ellipse the same will hold true It's just that the b square will get replaced with a minus b square. Okay. Now a Universal formula that I would like you to remember is eccentricity is always one plus semi conjugate axis length whole square by Semi transverse axis length Always it doesn't depend which case you are dealing with it's always one plus semi conjugate whole square by semi transverse whole square Right. So in this case your a becomes semi conjugate Okay, and b becomes your semi transverse Okay, in the previous case b was semi conjugate a was semi transverse. Okay Another important change that you will see is the length of the lattice rectum length of the lattice rectum in this case will become to a square by b Again the universal formula for this also is there universal formula for length of the lattice rectum is two times Semi conjugate the whole square by semi transverse value Okay, any questions So just like we had done a comparative study in case of the two standard forms of ellipses We'll also do the same for the two standard hyperbole. Let's go on to the next page so certain difference first of all in the equation is this equation will have a Structure like this and the conjugate of it will have a structure like this So where is the difference coming in the presence of one and minus one? In case of ellipses the difference was coming in which is more whether a is more or b is more, right? Here, there's no AB comparison here. The comparison is whether there's a one here or whether there's a minus one here Are you getting my point? Okay, so for such case your equation for such hyperbola your hyperbola will look like this I'm just drawing a rough diagram These will be your direct cases Okay, for such case your hyperbola will look like this Just a rough diagram So where will be the center? Where will be the focus? Where will the vertices? Let's look into that as well So, let me just name them first center. We'll find out. We'll talk about center. We'll talk about vertices We'll talk about eccentricity We'll talk about the four side We'll talk about the equation of the direct cases We'll talk about length of Lr lattice victim We'll talk about We'll talk about length of transverse axis and we'll talk about length of conjugate axis Okay So, let's do a comparative study quickly will not take much time here Right center for both of them will be zero zero center for both of them will be zero zero no difference in the center Okay, what is his hair will be a comma zero and minus a comma zero Whereas what is his hair will be zero B and zero minus B? Eccentricity here will be under root of one plus B square by a square Eccentricity here will be under root one plus a square by B square Okay, however, do not follow the do not forget the standard formula or the universal formula Forci here will be a comma zero and minus a comma zero Here the first I will be zero B and zero minus B The equation of the direct cases here will be x equal to a by and x equal to minus a by Here it will be y equal to B by and y equal to my Sorry y equal to minus B by Length of the latter system here will be to B square by here. It will be to a square by B Okay Length of the transverse axis here will be two way units This will be to be and this will be to be this will be to Is it fine? Let's quickly take a question. I Hope you have copied this down. These are the subtle changes Okay Let's take this question first This is just based on your basic locus definition of the hyperbola There's a hyperbola whose directrix is at 2x plus y equal to 1 focus is 1 comma 1 Remember whenever they mention a directrix and a focus it is the corresponding directrix They cannot mention your focus of one and the directrix of the other Okay, so this focus and these directrix are basically married Okay, and a senticity is root 3 which of the following gives you the equation of the hyperbola Let's have the pole One and a half minutes is good enough or two minutes max to max Time starts now one minute over. I have got one response only Okay, five four three two one What what what what? Okay, 14 of you 15 of you in fact have responded with most of you going for option B So as we discussed with you the locus of hyperbola is SP should be EPM Okay, so SP I'm directly using X and Y. Let's not waste time assuming h and k and then converting it to X and Y Save let's save our time and energy So this is our SP is Equal to E times which is root three times PM PM will be mod of 2x plus y minus 1 by under root of 5 Okay Remember most of you forget this root 5. Okay, so take this root 5 on the other side and square it Oh, sorry now, let's try to collect terms Here I will get 5x square But here I will get here. I will get 4 into 3 12 x square. Okay, so you'll have 7x square coming up. Okay Here I will get 5 Y square here. I will get 3 y square. So I'll get minus 2 y square I think clearly option number a comes out as the answer, but I'll still continue to solve this and Here I will get 4 x y into 12 which is 12 x y okay X term will be minus 4 minus 4x minus 12 x Sorry minus yeah minus 12 x and Here x terms Minus 12 and here it comes out to be minus 10. Right, so it'll give you 2x Sorry minus 2x minus 2x Oh, oh, that's where the answer differs So we should not be in a hurry to conclude Right Yeah, so minus 4x and this is 12x and here it is giving you minus 7x. It's minus 2x. Okay Y term will be minus 6 y and from here I will get so you'll have 4 y Okay, and this is going to be 210 and this is going to be 3 so minus 7 equal to 0 Which clearly means the answer is none of these The answer is none of these Okay, of course B is not correct But many of you went for D which was option Which was the right option. Okay, so D is correct in this Any questions anybody Any questions any concerns? Okay, let's move on to the next Find the foresight of this hyperbola very simple question. Let's have 45 seconds for this not more than that very good 5 4 3 2 1 Okay, most of you have gone with option number C Let's check So for finding the foresight the first thing is always we need to establish what kind of a hyperbola is this Okay So it is the hyperbola where you have the conjugate axis along y-axis and transverse axis along the x-axis. Am I right? Okay, so for such cases Semicity is under root of 1 plus B square by a square right B square by a square will give you a root 13 by 3 Okay Now 4 sign is that a comma 0 a is clearly 3 Okay, so 4 sign is root 13 comma 0 and minus root 13 comma 0 Option number C is the right option which most of you have gone with absolutely correct now When it comes to Shifted or generalized form. I don't think so. I need to discuss anything separately with you because it follows exactly the same way as what we did for ellipse and for a parabola as well So if let's say your center is shifted to let's say alpha beta If your center of a hyperbola is shifted to alpha beta it will become this or Depends upon whatever is the situation plus minus one I'm writing So I'm basically taking care of both the situation in one go Okay, and whatever you know about that corresponding hyperbola Just do a role change role play role change will help you to get the answer very quickly So I don't think so. I need to you know separately deal with this concept. We can rather spend time doing some questions Correct. Let's take few questions. Okay. Let's start with a very simple version Where do you think is the foresight of this hyperbola? Let the pole be on Let the pool begin one and a half minutes is good enough last 30 seconds Okay, Kinshuk noted five four three two One okay is the most Chosen option. So let's check whether he was correct or not So as you can see here that capital X is X only Capital Y is Y minus 2 A is for B is there. Okay, and the presence of a one Sorry presence of a one over here shows that it's a case where your Transverse is along x-axis or transverse is shifted Version of the x-axis. Okay So let's try to solve this So a centricity is what so first of all with without a centricity. We will not be able to find out the Forci coordinates. So it's under root of 1 plus B square by a square which is root 13 by 2 Okay, so for psi is x equal to plus minus ae and y equal to 0 So x equal to plus minus ae plus minus ae Oh my bad My bad. My bad. My bad. This is going to be four small judgment error. This is going to be five by four Yeah So plus minus Five Okay, and this is nothing but y minus two equal to zero Okay, so your answer is plus five comma two and minus five comma two Which is obviously option number a is that fine any questions? All right Let's take a question where we need to complete a square if at all we have any This we have already done Okay, let's take this question for a change Vertices of a hyperbola are at zero zero ten zero one of its four psi is at 18 zero Okay, find the equation of the hyperbola one and a half minutes Time starts now Okay, last 15 seconds five four three two one Okay, okay Most of you have gone with option number B Let's check So deal all I've done a rough diagram here. Please do not Mistake this to be your x-axis and this to be your y-axis It may be but I've not drawn it by that intention So I've basically shown you the vertices which is zero zero ten zero and I've shown you the one of the four side So first of all the center is the midpoint of the vertices. So this has to be five comma zero Right center is very important. So center is at five comma zero This distance is what we call as a and this distance is what we call as a e Okay, so a is clearly five and And a is clearly 13 that means he is 13 by five Right now in this case if you analyze it from your coordinate geometry point of view You are basically looking at a hyperbola, which is slightly shifted to the right by five units Correct. So it is a standard form which is shifted by five units to the right Correct, right So its structure will be somewhat of this nature X plus minus alpha whole square by a square minus y plus minus beta whole square by B square equal to 1 Why not minus 1? Because its transverse axis is either on the x-axis or parallel to it Okay had the transverse axis been along the y-axis or parallel to it Then I would have written a minus one over here. Oh, sorry. I wrote a minus one It's actually a one by mistake. I wrote a minus. Okay now In this case where your hyperbola was basically a case of transverse axis being You know parallel to the x-axis or you can say on the x-axis your B square was a square e square minus one so a is nothing but Fixed what about B square? Let's figure it out. So this is 13 square By 25 minus 1 this will come out to be 144 Correct center is at 5 comma 0. So alpha beta is 5 Alpha is 5 and beta is 0. So your final equation will become x minus 5 the whole square by 25 minus y square by 144 equal to 1 which is clearly which is clearly option number B Okay Now I will not take a lot of your time. I'll basically cover up two small concepts One is the concept of focal distance focal distances So if you have a hyperbola and let's say I take it as a standard case This is your S1. This is your S2 and this is a point x1 y1 Okay, what is the distance of this point from the two four sides? So what is S1 P and what is S2 P? Okay, I'll follow the same mechanism as what we did for What we did for an ellipse. So let's say I drop a perpendicular on the director which is M1 So S1 P is P M1 Okay, so what is P M1 P M1? This is nothing but x equal to a by so it is nothing but x1 minus a by So it gives you e x1 minus a Okay, similarly, this will be e times P M2. This will be a M2 Okay, and this will be x plus a by that is nothing but E x1 plus a Here you will see that the difference of the two focal distances is basically to a That means it is another definition or locus definition of a hyperbola Where it says that hyperbola is locus of a point Moving in a plane in such a way In such a way that the difference of its distances Difference of its distances from fixed points is A constant is a constant by the way many times in the books They will write the modulus here because you do not know which is more Okay, you do not know which is where whether S2 P is more or S1 P is more because if a point is here S1 P will be more Okay, and S2 P will be less. So in order to show that this difference is a constant We normally write a mod or sometimes in the books. You'll see they'll write S1 P difference S2 P is equal to 2. That's another definition of a of a hyperbola Okay, now here also a very important concept that comes out is if you Keep a light source as one of the focus after hitting the parabolic reflector it will take a path Which if you extend backwards will meet at the other focus I'll repeat If you keep a light source at one of the foresight After hitting the hyperbolic reflector, it will take a path which if extended backwards Will meet at the other focus Just like we had it for the case of a ellipse Getting my point okay, and Last thing that I would like to discuss with you is the parametric form and again parametric form is not going to be immediately Required, but I don't want to waste time when I am doing it with you in the After your semester exams classes So parametric form for let's say I take this standard case So there are actually two parametric forms that we normally talk about one is a seek theta and Y is beta and theta. This is one of the form which is mostly used In some cases you will realize that your books are also using This formula where theta here is a parameter. So this is another form But this is something which is very less seen Okay, I've seen this to be very less. How many times I'll also seen this kind of a formula a T plus 1 by T and Y is equal to B by 2 T minus 1 by T So these are different different, I know parametric forms, but the most commonly one uses this one, okay? Yes, absolutely. This is your Cosh Okay pronunciation is Cosh This is shine Okay So this is called hyperbolic functions. Okay, so X equal to a Cosh H theta you can write it and Y is equal to a shine It's not sign sign is if in shine shine The pronunciation is shine shine sign theta. Okay. Anyways If let's say if your hyperbola is the conjugate version that means if you have x square by a square Minus y square by B square equal to minus 1 We just flip the position. So basically here X will become a tan theta and Y will become BC theta Okay, so this is the most widely parametric form real. Why will I joke shine in Cosh? It's not a mispronunciation. It's the pronunciation that we use for hyperbolic function Cosh shine Hahaha Sounding funny, huh? Okay Let's do some questions Let's take this one. I Think simple one Yeah, yeah, there are so many parametric forms possible. It's up to you But the ones which are seen in the books are normally AC theta and a tan theta. Okay Let's have the poll for this our letter-sector length is 8 as intensity is 3 by root 5 What could be a possible equation for the hyperbola? C and D are definitely not hyperbola Actually, these chapters are very very long in hyperbola only we have a concept of rectangular hyperbola and all those things in fact, I can introduce it to you in one quick shot and After this problem, let's complete this Yeah five four One are quite mixed response a and b have got equal number of votes However, only eight of you have participated in this polling anyways So C and D cannot be your answer, right? And as you can see in your options They're basically talking about this kind of a hyperbola. Okay, so you have to work as per the options Okay, I know there could be two possible hyperbola, but work as per the options So as per this option your hyperbola's letter-sector is to b square by so this is given to you as 8 Ascenticity is given as 3 by root 5 Okay, that means b square is given as 4a Okay, b square is given as 4a and e square is nothing but 9 by 5 Now what is e square for such a hyperbola? It is 1 plus b square by a square Okay, that is 9 by 5 that means b square by a square is equal to 4 by 5 and b square is 4a right, so 4a by a square is 4 by 5 that clearly means a is 5 correct Yes, so if you see this it is actually x square by 25 minus y square by 20 equal to 1 and this is a case where your a is 5 so option number a has to be correct It cannot be b. Don't waste your time finding your b out because you'll not get a different answer from there Now one small thing I would like to talk about is rectangular hyperbola Just one minute of your time I will take So what is a rectangular hyperbola? It's a hyperbola where your length of the transverse axis is same as the length of the conjugate axis okay, in other words Instead of a b you can write a instead of this b you can actually write a a square So it becomes x square minus y square is equal to a square. Okay Now in school, they don't tell the actual picture. Why it is called a rectangular hyperbola. Does anybody know? Why rectangular hyperbola? Yes see We'll do it after the Semester exams in a hyperbola. There are actually asymptotes Okay, we'll talk about asymptotes. I should not draw it with this So asymptotes are basically lines which pass through the center and their tangents to both the arms Okay, these are called asymptotes asymptotes are line which basically touched the curve at infinity In a rectangular hyperbola, these asymptotes are at 90 degrees to each other Okay So when these asymptotes are at 90 degree a will automatically become equal to b because this line is actually x by a minus y by b equal to 1 sorry y y b equal to 0 and This line is x by a plus y by b equal to 0 So their slopes product is minus 1 slope here is you can take the slope to be b by a slope here is minus b by a So since m1 m2 is equal to minus 1 Minus b square by a square is minus 1 so a becomes equal to b Okay, that is why it is called a rectangular hyperbola But what is important for you is just to know that a is equal to b and a sent it a centricity for a rectangular hyperbola is a root 2 Okay So with this we Close the school level discussion of conic section next class will be on Probability, okay. That is also very easy chapter. Basically. We have introduction to probability. Thank you Take care. Bye. Bye. Stay safe