 Hello, welcome all to the YouTube live session and conic section ellipse So we're going to start with very very basic concepts of ellipse in today's class and Those in I would request you to please type in your name in the chat box so that I know who all are attending the session Hello shares. Hello to the guys will start with basic definition of an ellipse With the ellipse definition, so basically what is ellipse ellipse is and in terms Defined as It is defined as locus of a point which moves in a plane in Such a way its distance from a Fixed point which we know as focus from our experience from parabola Is a constant ratio is a constant ratio? From a fixed line from a fixed line that is the directrix That is the directrix and This ratio This ratio is called a centricity This ratio is called a centricity Denoted by e Whose value is between zero and one whose value is between zero and one, okay? So in this definition, it's the local point which moves in a plane Says that its distance from a fixed point is a constant ratio from a fixed line Okay, and this ratio is called the a centricity and the value of the a centricity in case of an ellipse is somewhere between zero and Definition I can actually derive the equation of an ellipse. So let us Let us take a fixed line Which we are calling as the directrix Okay, and let us take a fixed point which we are calling as the focus Okay now remember in case of an ellipse foci and Two directrises so we can have two foci S1 and S2 and we can have two directrises Says that the ratio of the distance of any point lying on the ellipse From the focus To that from the directrises is the same e So basically what I'm trying to say is Let's say I trace the focus of the part Particle or the moving point Okay, now this point is moving in this particular path in such a way that The perpendicular distance of this point From this electric, so that's a PM. This is that says one P. Okay So this ratio S1 P by PM is always e Where this e is somewhere between zero and one let's say you consider your e as point five Then wherever this point goes in This entire path the ratio of S1 E.5 right remember this directrix. Let's say D1 is Basically married to this focus S1 In a similar way this direct is D3 to this focus So as to take this point Let me just change the color of this Let's say I take the distance of this point from this focus S2 Then I have to take the distance from this directrix D2 that is Let's say I write it as M2. I write it as M1 So S2 P by P M2 will also be the same e So whatever you choose to draw the direct draw the ellipse the same e Would be true for this ratio as well as to P by PM So remember Whenever you're choosing the distance of P from S1 you have to refer to the distance of P from direct even Then only this ratio can be called as e So, please do not do that S2 P by P M1 this will not be e. Please note this please note this out Neither you should say S1 P by P M2 is e. So these two are very very important thing Only when it is taken from The corresponding focus and its corresponding directrix can we call this ratio as e else not Any questions so far guys, please feel free to type in all it Now in order to derive the equation, let us The standard equation of an ellipse Okay So let's now derive the standard equation of ellipse in order to derive the standard equation of an ellipse will consider We will consider the line joining the two focus the line joining the two force is your x-axis Okay, so what I will Consider this to be your x-axis becomes your x-axis okay, and The midpoint of the two foci. I will come the origin So if I drop a perpendicular line like this past I can easily call this as my y-axis Okay I'm considering this point that is the midpoint of the two foci to be the origin By the way a few terms that you must be Well aware of the midpoint of the two foci is called the center of the ellipse Okay, the midpoint of the two Forci is called the center of the ellipse so center is basically nothing, but It's the midpoint of s1 and s2 is the midpoint of the two foci s1 and s2. Okay now for the purpose of simplicity I'll call it as C comma 0 I'll call this points S1 as C comma 0 so s2 will become minus C comma 0 S2 will become minus C comma 0 Okay, let us call this point as A comma 0 let me call it as a and This point. Let me call it as minus a comma 0 Let me call this as a dash. Is that fine? All right now Stay very very carefully Now since can since a lies on the ellipse I can say that let me call this as and one. Let me call this as and two Okay since a Is on the ellipse the distance of a one distance of a distance of point a From s1 Divided by the distance of a from The directrix that is n1 Should be equal to the eccentricity of the hyperbola One right, which means a D now for the purpose of Simplicity, I will call this line as a equal to d line. Sorry x equal to d line So a minus c is this distance Okay, divided by d minus a should be equal to the eccentricity of the Parabola ellipse, so let me call it as the first equation and Let me call this equation as x equal to minus d So I'll repeat once again what I have done. I've taken the directrix equation d1 to be x equal to d and Directrix equation d2 to be x equal to minus d I've taken the coordinates of the point a to b a comma zero I've taken the coordinates of a dash to be minus a comma zero I have taken s1 to be c comma zero so s2 becomes minus c comma zero okay Because a point lies on the ellipse. Can I say it's distance from s1? divided by Let me write it as a s1 a s1 by a n1 By a n1 will be equal to e so I get this Okay in a similar way, can I say a s2 by a n2 Will also be now what I'm doing is I'm considering the distance of a from this focus s2 divided by distance of a from the other directrix so a s2 by a n2 will also be which implies a plus c a plus c and This distance will be a plus d This is since will also be equal to e. Let me call it as a second equation. Okay Now you must be one Number one in equation number two because Unnecessarily there are too many variables getting involved in this particular situation like see a E so what I want to do. I want to minimize the number of variables being used Okay, so I want to write my c and d in terms of a and e only so I want to take a and e as Our you know We should be appearing everywhere. I don't Necessary c and d to come up So from equation number one and two I want to eliminate c and d or I want to get what is the value of c and d in terms of a and e only Okay So let us solve this. So from one I can say a minus c is ed minus ea and From two I can say a plus c is equal to Ed plus ea Let me add it So two a is equal to two ed So this implies d is equal to a by e see Now I'm able to get my d in terms of a and e so I don't want to use too many variables in my given scenario I want to reduce the number of variables used Now if you subtract it that means to see Will be equal to two a e so c will become a e Correct. So I've now got my focus c in terms of a and e So this line that you see over here This line that you see over here instead of writing x equal to d you can also write it as x equal to a by e and Instead of writing x equal to minus d you could write it as x equal to minus a by Yes, there's nothing wrong in using c and e but you don't want to use so many names c, d, a etc So it'll be too much to handle So similarly the focus could be written as a e comma zero and this focus could be written as minus a e comma zero Is that fine? Now I've still not derived the equation of the ellipse. I'm just Reducing the number of variables used. So what I'll do is I'll again quickly draw the ellipse over here okay, and let's say these are the two directories and This is my y-axis. This is my x-axis okay Now for the purpose of you know reusing this I will keep It's a snapshot so that you know, we don't have to redraw it every time Now once again, this is a this is a dash We consider this coordinate to be a comma zero this to be minus a comma zero This is your s1 s2 s1 was now a comma zero and minus a comma zero The direct x d1 has the equation x equal to a And this has the equation x equal to minus a by Remember in n crt. They write c as a Right so remember still c is equal to a but I prefer not using c but using a e in terms of c, okay? However, it's completely your own how however, you are can use it which is a zero comma zero now Start the finding the equation of an ellipse Which is oriented in this manner that its center is at origin and It's accesses are along the x and the y-axis so before we start guys two terms that we need to learn from it This is called the major axis This is called the major axis of the ellipse because The reason why this called the major axis is because this length a a dash That's more than this length which we call as the BB dash So this will be called as the minor axis. So there are two accesses Remember there was only one axis in Parabola which used to divide it into two symmetric halves in case of a ellipse You will have two accesses one is the major axis and Another one is a minor axis again Listen to two equal halves Then what major is because it is of a longer length than BB dash So BB D B dash is of a smaller length B dash And of a bigger length, okay so, uh Let me tell you that a a dash is called the length of the major axis Length of the major axis and BB dash is called the length of the minor axis It's called the length of the minor axis now so let us start the process of Finding the equation by assuming a point h comma k on the ellipse So according to the definition of the ellipse the distance of this point p from the One of the focus s1 and the director's corresponding to it Should be equal to e s1 p by pe1 should be equal to e So we'll use this locus definition to get the equation of the standard ellipse So s1 p will be nothing but Under root of h minus ae whole square Plus k minus zero whole square and p m1 will be nothing but mod a by e minus h And this ratio should be equal to e as per the definition of an ellipse Okay, so let me further simplify it first. So let me take it on the other side. So under root of H minus ae square plus k square is e times mod a by e minus h Okay, which is nothing but a minus e h mod. So let's square both the sides So i'm just directly jumping to the squaring of both the sides Okay, which will give you the expression as h square plus a square e square minus 2 h ae Plus k square Is equal to a square Plus e square h square minus 2 h ae. So this will be cancelled off Okay And let me bring this term to the other side. Let me bring this term to the other side So it'll become h square 1 minus e square plus k square and let me bring this term to the other side So it becomes a square minus a square e square Okay So further simplifying this you can write it like this equal to a square 1 minus e square Now divide both sides by a square 1 minus e square So by doing that you get h square by a square Plus k square a square 1 minus e square equal to 1 Equal to 1 so this term over here when you generalize this When you generalize this you get the required equation of the ellipse that is x square by a square plus y square by a square 1 minus e square equal to 1 now guys for the purpose of convenience we Call this a square 1 minus e square term as b square Okay, we call this term as b square Which makes the equation of which makes the equation of the ellipse finally to be this Which is basically known as the standard form of the equation of the ellipse So this is one of the standard form of the equation of the ellipse Okay, is there any questions so far? With respect to this derivation Now if you see this term carefully it is basically nothing but a square minus a square e square Which normally in your ncrt textbook can also write it as a square minus c square is b square Which means c square is actually c square is actually a square minus b square Okay, so this probably what is there in your ncrt textbook But uh I normally do not like using c. I use mostly a and e combination everywhere Now from this formula, you can also get the eccentricity e in terms of a and b by the way Let me show you what is b for you in the diagram Once we have got that the equation of this ellipse is x square plus x square by a square plus y square by b square equal to You realize that b corresponds to the y coordinate of the point where the ellipse cuts the y axis So if you make x as zero You realize that y square is equal to b square. So y becomes plus minus b So this point is actually 0 comma b and this point is actually 0 comma minus b okay, so 0 comma b and 0 comma minus b represents now in this case The end points of the minor axis Is that okay? So you can say length of the major axis is to a And length of the minor axis is now going to be to b So a dash which is equal to the difference distance between these two points is your length of the major axis b b dash which is Basically the distance between these two points is the length of the minor axis which is equal to 2b. Okay So guys a few analysis that I would like to do here If you see the expression of b square, it's actually a square 1 minus e square fine That means I could write 1 minus e square as b square by a square Which means I could write e square as 1 minus b square by a square which means I could write my e as Under root of 1 minus b square by a square Okay, so this gives you a formula for finding the eccentricity of the ellipse given that you know its Length of the minor and the major axis So you can also write this as under root You can also write this as under root 1 minus The semi minor axis length Whole square divided by the semi major axis the whole square Why I'm calling it as semi because b is actually half of this length, right? so b is actually Your length of the semi minor axis semi means half Okay, it's the length of the semi minor axis Okay And a is basically nothing but the length of the semi minor axis semi major axis length of the semi major axis Okay So please note down this formula a very very important formula for finding the eccentricity of an ellipse when you know the length of its Major or minor axis or for that mine for that matter the length of the semi major or semi minor axis Is that fine guys? Any questions? Please feel free to type in in the chat box if you have any questions Now guys few things that again I need to talk about from this equation Since we know that e square is greater than zero Okay e square is greater than zero I can say 1 minus b square by a square would be greater than zero Correct, which means that one is greater than b square by a square Which means a square is greater than b square Yeah, that means a is greater than b Right, that's why 2a is greater than 2b That's why 2a is greater than 2b. That's why the name of this axis was given as the major axis Because its length is greater than the length of the minor axis Right, which is 2b. So I hope guys this is pretty clear in your mind. How is the equation derived first of all What is the eccentricity of the ellipse in terms of the length of the semi minor and semi major axis? What is the relevance of the point b comma 0 comma b 0 comma minus b Etc Now guys remember that this was one of the standard cases So The ellipse that you just now saw was one of the standard cases So this is just one of the standard cases where Your x axis is the major axis y axis is the minor axis Okay, you can also have a situation where You can also have a situation where The ellipse instead of being Placed like this is actually placed like this Where this is these two are your directresses Let me just name it bb dash aa dash Remember Here your a a dash was your vertices. So a a dash A and a dash are called the vertices Are the vertices Okay, here b b b and b dash are called the vertices Okay Here we had the foci of the ellipse on the x axis which is a comma 0 And minus a comma 0 You can also have the foci On the y axis remember foci is always on the major axis Please remember this i'm writing down over here Please remember the foci and the vertices are always On the major axis Are always on the major axis So in this case, uh, let me call this as a comma 0 point and minus a comma 0 point This is 0 comma b. This is also 0 comma b And this point is 0 comma b e this point is 0 comma minus b e Let me write it over here And again, there'll be change in the equation of there'll be a change in the equation of The directresses as well There'll be a change in the equation of the directresses here. The directresses was x equal to a by e And x equal to minus a by e Here it'll be y equal to b by e and y is equal to minus b by e So there are two standard cases one is this one which we already derived and this is the second one Okay So second one also if you go on to derive the equation you will get a similar equation In fact, we'll get the same equation for both of them x square by a square plus y square by b square equal to one here also You'll get x square by a square plus y square by b square equal to one The only difference in these two equation is here a will be greater than b Here b will be greater than a so remember I'll just do a comparison of these two, uh ellipses Here your eccentricity is under root one minus b square by a square Before I move on to eccentricity, let me tell you about the major and minor accesses So here your a a dash is your length of the major axis Length of the major axis Here your b b dash is the length of your major axis Okay, here b b dash is the length of your minor axis Okay, here a a dash is the length of your minor axis By the way, these lengths are respectively 2a and 2b and in this case These lengths are 2b and 2a So just understand this comparison which I'm drawing between these two ellipses Next is eccentricity which we already just saw was actually this formula For this case of an ellipse, but for this case of an ellipse, it will be one minus a square by b square Okay, however guys remember that whether you are looking at Case one or case two of the ellipse your eccentricity will always be One minus semi minor axis By semi major axis the whole square always So this formula is applicable for both the scenarios This formula is applicable for both the scenarios Okay So even if the situation changes This formula will still remain the same. So just remember this formula Okay now There is something which we also studied in case of a parabola which we call as the lattice rectum Okay, so what is the lattice rectum? Lattice rectum is a focal chord Which is also a double ordinate, right So there are two lattice recta There are two lattice recta in case of an ellipse. So as you can see In situation number one, these are the two lattice recta Okay And in case of situation number two ellipse Both are standard forms, but in the second standard form. This is your lattice recta Okay So let us now focus on finding the length of the lattice rectum for case one and case two Okay Now it's going to be easy because we just have to find out the difference of the y coordinates of these two points Let me call them as p and q Okay, so remember p Will have the same x coordinate as the focus I'm not sure about the y coordinate that we have to find it out Even q will have the same x coordinate as the focus and y coordinate. We need to find it out And the difference of these two y coordinates will give us the length of the lattice rectum So let us find it out So p has let's say a comma y Now this satisfies the equation of the ellipse. This must satisfy the equation of the ellipse Correct So, let me put let me put x as a e so a square e square by a square minus y square b square equal to 1 So a square a square gets cancelled Okay So I get y square by b square is equal to 1 minus e square Are there no problem? You can start watching from you know the beginning Okay now I can write y square as b square 1 minus e square Okay, now remember remember Your e square was 1 minus b square by a square So 1 minus e square will be b square by a square So substitute it over here Substitute it over here You would realize that your y square will become b square into b square by a square That is nothing but b to the power 4 by a square. That means your y is plus minus b square by a So going back to this point over here Now I know the coordinates of these two points Now I know the coordinates of these two points Which is going to be b square by a and minus b square by a So the length pq So the length pq The length pq is going to be 2b square by a so this is very very important. This is the length of the lattice rectum 2b square by a is the length of the lattice rectum for the case 1 Okay, similarly for case 2 you can derive that Your length of the lattice rectum will be 2a square by b 2a square by b units So what I will do next is I will Clearly chart out the difference between these two cases of the ellipses And then we'll concretize the concept by solving few problems so that you are clear absolutely with the basic idea about How these two ellipse, you know are positioned What are their critical points and what are their critical equations? Okay So please look at this slide very very carefully This slide I will be talking about comparison Comparison Between two standard forms Between two standard forms of ellipse Okay So both the forms Equation looks exactly the same, but there is a subtle difference The difference is here a is greater than b here b will be greater than a Okay, figure wise I'm just drawing a miniature version so that you know we can use the entire screen figure wise it looks like this And this fellow figure wise will look like this So you can call it as like an egg lying on the ground. This is an egg lying on its nose Okay, so you have seen an egg. So if you just rest it on the ground, this is how it will rest but if you uh, you know Make it stand up on its nose. It will look like this Okay, so please be very very careful regarding the differences between these two ellipse So let me draw this separation line Next is vertices Uh, sorry center Let's start with center first Both the centers Would be at zero zero. So this also your center would be at zero zero Vertices for this ellipse will be at these two points a and a dash So a is A comma zero and a dash is Minus a comma zero In this case your vertices will be at b and b dash So b will be zero comma b And b dash will be zero comma minus b Third difference is the position of their foci Let me call it as s1 s2 Here also, let me call it as s1 s2 So for foci s1 is A e comma zero And s2 is Minus a e comma zero In your school you would be writing it as c comma zero minus c comma zero, right? Here the foci would be at s1 which is zero comma b And s2 which is zero comma minus b Next equation of the directories Equation of the directories In this case it will be x equal to a by e and x equal to minus a by e Whereas in this case the equation of the directories will be y equal to b by e And y is equal to minus b by e I think I have shown you in the previous figure as well Yeah, y equal to b by e and y equal to minus b by e, okay Next is length of major axis Length of major axis That's 2a Here Length of major axis Is 2b You can write 2b units length of minor axis Is 2b Here length of minor axis Is 2a Equation of major axis That's your x axis You can write it as y equal to zero Okay In this case Equation of major axis Is x equal to zero that is your y axis Equation of minor axis in this case Is your x equal to zero or y axis In this case equation of minor axis Is y equal to zero which is your x axis Ninth difference or ninth comparison Ascenticity is 1 minus b square by a square In this case it is under root of 1 minus a square by b square In this case length of lattice rectum Length of lattice rectum Is 2b square by a units And in this case length of lattice rectum Is 2a square by b units So please remember this chart because it is going to be very helpful At least in your school exams to solve a lot of basic problems Okay So I hope this comparison chart is clear in your mind There is no doubt whatsoever regarding this comparison chart So let's take some basic problems On whatever we have done so far Find one the center Second vertices Third for Ascenticity Fourth the foci Fifth Equation of directories Sixth length of major axis Seventh length of minor axis And eighth the length of lattice rectum Length of lattice recta I should say Length of lattice rectum for For x square by nine Plus y square by four equal to One And second question is x square by 16 plus y square by 25 equal to one So please work on the first one we'll discuss in a minute's time All right, so I'm assuming that you guys have done the first one Have you found out all these seven eight parameters for the first ellipse? Please type yes on your screen if you have done that so that I can start the discussion All right Only share says yes Okay, so let's discuss Now from the very Look and feel of it. This is your a square and this is your b square and you can clearly see that a square is greater than b square Or your a is greater than b So this is a case where your ellipse is basically like this The horizontal one Right Now try to recall everything with respect to this So we know that center is going to be Zero zero no doubt Correct Vertices are at a comma zero And minus a comma zero So that's going to be three comma zero and minus three comma zero Because your a is going to be Three and b is equal to two When it comes to eccentricity eccentricity will be under root of one minus b square by a square which is four by nine So which will be under root five by three Okay For sigh will be a e comma zero That is root five comma zero And minus a comma zero which is minus root five comma zero Equation of the directresses will be x equal to a by e x equal to a by e And x equal to minus a by e Okay Length of the major axis will be two a which is six units Length of the minor axis will be two b which is four units Length of the latter sector will be two b square Two b square by a which is going to be eight by three units So for the first question, I hope you have got these answers How many if you got all of them correct? How many if you got all the all of the eight parameters correct for the first one Great. Yes. Great. Okay So, let's now move on to the second one. So I'll just erase it to make some more space Okay So for the second one as you can see your a square is 16 and your b square is 25 That means your a square is less than b square. That means if a is less than b So it clearly is a case where your ellipse is the one which is The vertical ellipse the one which is standing on its nose. Okay like this. Okay Okay So just try to recall everything with respect to the second standard form which we had discussed a little while ago Remember center will still be at zero zero. So there'll be no change in the center Okay, vertices will be zero comma b and zero comma minus b. So zero comma five And zero comma minus five. Okay Ascentricity will be under root one minus Semi-minor semi-minor here will be a this is your semi-minor a sorry Semi-minor which is a And this is your semi major b. So it'll be one minus a square by b square It'll be one minus a square by b square, which is going to be nine by root nine by 25, which is three by five Okay Fosai will be zero comma b e And zero comma minus b. So it'll be zero comma three And zero comma minus three. Okay Equation of the directories will be y equal to b by e y is equal to b by e remember b is five And e is three by five So it'll be 25 by three and y is equal to minus 25 by three. Okay Length of the major axis will be two b Two b, which is actually two into five. That is 10 units Length of the minor axis will be two a which is two into four, which is going to be eight units Length of the latter sector will be two a square by b which is two into 16 by five, which is 32 by five units So this is the answer for the second question. How many of you got All these eight parameters correct even for the case of the second ellipse. Okay, great So, let me ask you one more question before we move on. I'll just ask you one directed question Find Find Equation of directories for x square for x square plus y square Equal to nine and please feel free. Please type in the response in the chat box Plus minus two root three. Okay. Let's discuss this So first of all it is not written in the standard form So I can write it as x square Four by nine y square by nine equal to one Right, which can further be written as x square by nine by four y square by nine equal to one Okay, which is x square divided by three by two whole square Plus y by three whole square equal to one Okay, now clearly here a is less than b right because a is three by two and b is three That means it's a case of A ellipse which is like this which is standing on its nose like this right So here the equation of the directories would be y equal to b by e and y equal to minus b by e So for that we need to know our b and e clearly So b will clearly be three And e will be under root of one minus a square by b square. Remember here semi minor axis is a semi major axis is b so one minus nine by four divided by nine which is going to be Three by four under root which is root three by two Okay, so y is equal to three divided by root three into two And y is equal to minus Three divided by root three by two So that's going to give you the answer as y equal to two root three And y equal to minus two root three as your equation of the directories. Please do not forget to mention why You're not mentioning any number. You are mentioning an equation. So variables should not be missed out Great. So we'll take a few more questions before we move on to other concepts This question is based on the formation of the equation of an ellipse find the Find the equation of an ellipse Find the equation of an ellipse Who's One of the four side Or one of the focus is minus one comma one Ascenticity is half and directrix equation is And directrix equation is x minus y plus three equal to zero You can take the pic and send it to me on my personal whatsapp chat So guys again here we'll be using So this is a line like this Uh focus is given to us as minus one comma one. So let's take it like this minus one comma one So It will be an ellipse of this nature remember Major axis will always be perpendicular to the directrix Major axis will always be perpendicular to the directrix. Okay. Anyways So we'll just have to use the basic definition of an ellipse that it's distance of any point from this fixed point That is minus one comma one To that from this fixed line, which is your directrix That ratio will always be half So I can say under root of x plus one whole square y minus one whole square is equal to e square e times Or this divided by This will be equal to e remember This is s m this is sp and this is p m For p m. I have used the distance of the point x comma y from the line Okay So under root of x plus one the whole square Plus y minus one the whole square is e times this Let us square both the sides If we square both the sides, this is what we get Okay So it becomes eight times x square plus y square plus 2x minus 2 y plus 2 Is equal to on the right hand side. We'll get x square plus y square plus nine minus 2 x y plus 6 x minus 6 y And if you simplify this You are going to get this as the answer 7 x square 7 y square Plus 2 x y 16 x minus 6 x which is 10 x minus 16 y plus 6 y Which is minus n y And 16 minus 9 which is plus 7 equal to 0 So do you realize how complicated the equation of An ellipse may be which depends upon Which depends upon the situation given to you Okay Good. So some of you have solved it absolutely correctly. So we'll move on to the next question Next question is find the equation of the ellipse Find the equation of the ellipse of the ellipse Whose vertices whose vertices is 0 comma plus minus 10 and eccentricity is 4 by 5 Whose vertices is at 0 comma plus minus 10 And eccentricity is 4 by 5 Aker's slight mistake in your answer Hope you would have got the mistake Absolutely correct. Absolutely correct. Okay. So the moment you actually plot these vertices you would realize that it's a case of a Vertical ellipse like this It's a case of a vertical ellipse Of this nature. So this is 0 comma 10 And this point is 0 comma minus 10 So, you know your ellipse is going to be of this nature x square by a square plus y square by b square equal to 1 Where b is equal to 10 Where b is equal to 10 Because this length is your b This length is your b Okay, and this length is your a So, how do I get a from this? By the using the simple fact that in this case e will be under root 1 minus a square by b square Right So e is 4 by 5. So let me replace this with 4 by 5 So it implies 16 by 25 is 1 minus a square by 100 Which means 9 by 25 is a square by 100 Which means a square is equal to 36 Which means your answer will be x square by 36 plus y square by 100 equal to 1. So this is your answer Well done guys to those who answered Absolutely correct. No doubt about that Now we'll move on to the case of generalized form of equation of ellipse Now so far whatever we saw was the case of standard form of the equation of an ellipse where The center of those ellipse were at origin And the measure in the minor axis is where exactly on the x on the y axis is okay But let us assume a case where your ellipse Center is not at the origin. For example, it is somewhat like this It's somewhat like this So center has now come to a point. Let's say alpha comma beta Okay center has now come to a point alpha comma beta And this is your New major axis. This is your new minor axis Okay, but however remember even though the center has gone to alpha comma beta Your new major axis is parallel to the old major axis And your new minor axis is parallel to your Old minor axis okay So how in this case the equation of the ellipse is going to change the equation of the ellipse is now going to just undergo a small change Instead of just x square by a square it will now become x minus alpha whole square by a square And this will become y minus beta whole square by b square equal to one Right. That is what I already discussed with you in your school. You you are taught that a is equal to c And that's how we get c as a and I normally use a instead of c I only deal with a b e that's it See guys, it doesn't matter whichever approach you follow. Just stick to it. Ultimately your answer would be the same So now we'll take up some questions with respect to Such generalized form or we call it as a shifted form of an ellipse Okay So first of all, I'll start with giving you a question Where I would mention you the equation of the ellipse and ask you to find the critical points or equations So let's start with this question Find center vertices 4 psi equation of the directories Okay, uh, let's say Length of the lattice rectum For this ellipse Let's say I give you an ellipse, uh x minus 2 whole square by 9 y plus 1 whole square by 16 equal to 1 Exactly exactly divi we can get it from there also Yeah, same thing same thing. There's no difference between both the this thing answers Approach may be slightly different, but there's no difference in the final answer In case of n crt, they switch the position of a and b But in reality you are actually dealing with numbers Right, so irrespective of whatever approach you follow the answer would be independent of that In school they teach you x square by b square plus y square by a square equal to 1 But what I do I keep a and b same I just say a is greater than b in one of the cases and in other case b is greater than a Yeah, yeah, absolutely whichever method you follow. It's going to be the same answer What method I am following is the same given in adi sharma and you'll also find it in many other books The approach of n crt is very much, you know specific to n crt only Now guys, uh, when you're solving such kind of problems, okay, just again do a role play this resembles This resembles This form Okay, and since your b is greater than a it's the case of a vertical ellipse right now Just do a role change So as to say that your capital x is equivalent to x minus 2 capital y is equivalent to y plus 1 And of course a is 3 b is 4 Okay Now when you're asked the center just try to recall everything which you have learned about This ellipse apply it to this ellipse similarly So for this ellipse, we know that For this ellipse, we know that Your center is that 0 comma 0 right So instead of writing 0 comma 0 write it as x equal to 0 and capital y equal to 0 So here you just do a role change your capital x will now become x minus 2 And your capital y will become y plus 1 So which means x will be 2 y will be minus 1 So your center will be at 2 comma minus 1 next Next we are asked to find out the vertices For this kind of For this kind of ellipse where b is greater than a we know vertices is that 0 comma b And 0 comma minus b So just do the same role change over here x minus 2 is 0 And y plus 1 is equal to b And again x equal to 0 y plus 1 is equal to minus of b Which means This will give you x as 2 y as 3 That means 2 comma 3 will be your one of the vertices This will give you x as 2 and y as minus 5 Which implies 2 comma minus 5 will be the vertex Is that fine? Now ascenticity let's find ascenticity first before we find the focus Ascenticity will be Ascenticity will be Under root 1 minus a square by b square Okay Which is going to be under root of a square which is equal to 9 b square which is 16 that's going to be root 7 by 4 Okay So now let's focus on the eccentricity. Sorry. Let's focus on the foci So foci coordinates is x equal to 0 y equal to b e And x equal to 0 y is equal to minus b So if you do a roll change If you do a roll change it will become x minus 2 equal to 0 and y plus 1 is equal to b e b e will be root 7 And similarly y plus 1 equal to minus root 7 Okay Which will give you x equal to 2 and y equal to root 7 minus 1 And this will give you x equal to 2 and y equal to minus root 7 minus 1 So your 2 foci is 2 comma root 7 minus 1 and 2 comma minus root 7 minus 1 Is that clear? Similarly, what do we want next? We want equation of the directresses Equation of the directresses. Let me do it over here equation of the directresses For this equation is y equal to b by e and y is equal to Minus b by e right? So try to do a roll change When you do a roll change, it will become y plus 1 is equal to b by e b by e is 4 divided by root 7 by 4 That means it will become y plus 1 is equal to 16 by root 7 And y plus 1 is equal to minus 16 by root 7. So these two are the equation of the directresses Okay Length of the lattice rectum is 2 a square by b. Remember, it's a case of a vertical ellipse because b is greater than a So in this case, it will be 2 a square That is 9 by 16 Sorry 9 square by 4 by 4 not 16. Yeah by 4 That's going to be 9 by 2 units 9 by 2 units So guys, hope you are clear with how to find the critical points and the critical equations when your Ellipse is the generalized form of an ellipse So now given that you have now learned this particular Type of problem. I'm not going now going to give you a similar one find e 4 psi and length of the lattice rectum for x minus 1 whole square by 4 y plus 1 whole square by 1 equal to 1 Yeah, sure did it. I've already given to you So share is saying uh assenticity is root 3 by 4 Okay, you mean root 3 by 2 So please please, uh, you know mention your answer in the chat box. So e what is e? What are the coordinates of the uh Focus Yeah, that's correct. Yeah, that's correct assent. Assenticity is correct. Yeah, again just a comparison that you have to do compare it with x square by a square y square by b square equal to 1 where a is greater than b than b Okay, so assenticity for such cases where a is greater than b is under root 1 minus b square by a square So it'll be under root 1 minus b square by a square is 1 by 4 Which is actually under root of 3 by 4 which is root 3 by correct Now focus So focus for such cases will be x equal to ae y equal to 0 or you can say plus minus ae and y equal to 0 So x will be x minus 1 plus minus ae ae will be 2 into root 3 by 2 And y equal to 0 means this equal to 0 which means x can be plus minus root 3 plus 1 and y is minus 1 So your foci will be foci will be root 3 plus 1 comma minus 1 and minus root 3 plus 1 comma minus 1 So this will be a foci now What is your lattice sector for such cases? Lattice sector for such cases is 2 b square by a 2 b square by a units correct. So which is going to be 2 b square by a which is going to be 1 L r is going to be 1 1 unit Excellent. So guys are quite familiar with the generalized form as well So we'll take one more example and then I'll give you a break So one more question and then we'll take a break find eccentricity foci length of the lattice rectum for 4x square plus y square minus 8x Plus 2 y plus 1 equal to 0 Now instead of giving you in a very very clear cut fashion of x by a square plus y by b square I have given to you in a Expanded fashion, okay So find e foci length of the lattice rectum for this case absolutely correct Gaurav So first you need to convert it to a standard form and then start finding out the critical points or whatever other Parameters given to you in the question Please type done if you're done And do also post your answer whatever you have got So how do we solve these kind of problems? We first group up your x terms together y terms together Okay Uh, this is nothing but four times x square minus 2x This is nothing but y plus 1 whole square So this we can write it as x square minus 2x plus 1 and since you added a 4 over here, you have to add a 4 over here as well correct So it's 4 times x minus 1 whole square y plus 1 whole square is equal to 4 So it actually boils down to this equation Yeah, yeah, yeah, correct correct rm on so now this is going to be a vertical ellipse Right. So eccentricity is going to be under root 1 minus a square by b square a square by b square Is going to be this again root 3 by 2 is the eccentricity So this is done What about the foci foci will be at 0 and y equal to be And it would be at 0 y equal to minus of b so x minus 1 will be 0 And y plus 1 will be b e So b will be root 3 And similarly here y plus 1 will be negative root 3 So your foci would be at 1 comma root 3 minus 1 and 1 comma minus root 3 minus 1 Okay Latus rectum will be 2 a square by b 2 a square by b which is 2 into 1 by 2 it's the exactly the same thing I guess the same question But written in a different format Okay So now you can enjoy yourself a break. I will resume at We'll resume at uh 5 58 p.m All right guys, so let's resume for maybe left off Now we have already learned the equation of the ellipse in the standard form and the equation of the ellipse in the generalized form so Now the time has come that we learned the equation of an ellipse in a parametric form Equation of an ellipse in a parametric form However, this concept is over and above your school level syllabus Okay So when we talk about the equation of an ellipse in the standard form x square by a square plus y square by b square equal to 1 We can write down the same equation as x equal to a cos theta And y equal to b sin theta So this equation will be called as the parametric form for this equation So this is the Cartesian form This is the Cartesian form of the equation of an ellipse and this is called the parametric form This is called the parametric form Okay Where theta is a parameter Where theta is a parameter right Now taking a clue from this could you suggest me a parametric form for Such an ellipse which is generalized version of an ellipse x minus alpha square by a square plus y minus beta square by b square equal to 1 Can somebody suggest me a generalized? Can you can somebody suggest me a parametric form for this ellipse? What should I write x as and what should I write y as? Exactly absolutely correct the wave alpha plus a cos theta And this I can write it as beta plus b sin theta So this is the parametric form for the same ellipse which we had done a little while ago Okay Next concept that we talk about is the concept of focal distances of a point focal distances Of a point so let's take a Let's take a standard ellipse x square by a square plus y square by b square equal to 1 okay And if I give you any point let's say x 1 comma y 1 then the distance of Then the distance of p from s 1 And the distance of p from s 2 they are called the focal distances. So s 1 p and s 2 p would be called as The focal distances would be called as the focal distances. So Can you give me s 1 p value and s 2 p value in a simplified form? Can you guys find s 1 p and s 2 p in a simplified form? now, please note that Many of you would be applying distance formula but remember you just use the form that The locus of an ellipse is Nothing, but every points ratio from a fixed point to that from a fixed line is e So use this fact that s 1 p Is your e times p m 1? and s 2 p s 2 p is e times p m 2 Okay, that would be faster way to solve this problem So s 1 e Sorry s 1 p could be written as e times P m 1 p m 1 you can write it as this is your x equal to a by e, right? So a by e minus x 1 So when you open it up it becomes a minus e x 1 Okay Similarly your p m 2 p m 2 would be A by e plus x 1 that will be a plus e x 1 Okay, so just remember the distance of a point x 1 y 1 from Uh, the nearest focus is going to be a minus e x 1 and from the further focus is a plus e x 1, okay A very very important concept comes out from here that the sum of the two focal distances The sum of the two focal distances Will actually be a constant and that's actually the length of the major axis So this is a very very important definition That in case of any ellipse The sum of the focal distances of any point from two fixed points Is always a constant and that constant is equal to two a That means if you take two points And you connect them by a string You connect them by let's say a string Of length to a Okay, and then you take a pen And then you take a pen like this and keep this string taught Right these two strings should be taught And start rotating your pen You would realize that When you're rotating this pen You would automatically trace an ellipse Okay So this length that is s S 1 p plus s 2 p will always be equal to the length of this string So this will always be equal to s 1 p plus s 2 p and the Mark that you will make on the plane will be that of an ellipse if you keep these two ends of the string as taught So on basis of this, let's take a small question A man running A man running a race A man running a race Notice that the sum of the distances The sum of the distances from two flag From two flag posts Is always 10 meters Is always 10 meters and the distance between the flag posts And the distance between the flag posts Is 8 meters So just the equation of the path traced by the man Suggest A equation Of the path traced by the man Question of the path traced by the man Yeah, correct. Correct. Akash. So the sum of the distances is basically 2a. So 2a is 10 So you can say a is 5 And the distance between the two flag posts is actually the distance between the two four sides Which you can say is 8 That means your a is equal to 4. That means your e is equal to 4 by 5 So if e is equal to 4 by 5, you can say e square, which is actually 1 minus b square by a square So your b square will become a square 1 minus e square, which is 25 1 minus 16 by 25 That is going to give you 9 So you're going to get x square by 25 plus y square by 9 equal to 1 So let us take The next concept here Just like we took some extra concept in case of parabola and circle. Here also we'll be doing some extra concepts Uh, but before that I would like to do some problems with you Okay If the angle Oh, just let me once again, uh, share it with you. I think If the angle Between the straight lines Guys, is it visible now or still no screens? But from my end, it says it is shared. Yeah, it's shared Just let me, uh Refresh it Guys, I'll I'll end this session and I'll send you a new invite Because I think some problem has kept in with this