 Hello, all welcome to the YouTube live session on Koenig section So this class will be focused on ellipse the advanced level concepts the J main and J advanced level concepts So those who have joined in the session I request you all to type in your names in the chat box So guys, I'm going to start this session with a quick recap on the Standard form of the equation of the ellipse. So these are the two standard forms that we normally deal with Okay, so these are the two standard forms of the ellipse So when I say standard forms means those ellipses Whose center is at origin and whose major and the minor accesses are aligned along the X or the Y accesses Okay, so just a quick recap of the standard form of an ellipse So we deal with two types of standard form one is X square by a square plus Y square by B square equal to 1 where a is greater than B Right as you can see a is greater than B and the other one is where B is greater than a And the other one is where B is greater than a The difference in these two ellipse is basically in their orientation if you look at this kind of an ellipse it is basically an ellipse which is of this nature Okay So it's a major axis is along the X axis and the minor axis is along the Y axis Okay, and this kind of ellipse is where your orientation of the ellipse is in Such a way that your major axis is along the Y axis and your minor axis is along the X axis. So this is your Y and this is your X Okay So this is a a dash B B dash a a dash B B dash So a few things about both the ellipses that we need to keep in our mind first The center for both the ellipses is origin 0 comma 0 The vertices a and a dash is called the vertices. So for this ellipse, let me call this ellipse of type 1 or Standard form 1 and this is standard form 2 for standard form 1 the vertices are at a plus minus a comma 0 whereas for standard form 2 It is at 0 comma plus minus B The length of the major axis or you can say the length of the semi major axis is going to be a in this case Whereas the length of the semi major axis in case of the type 2 standard form is B Okay, foci for this is remember foci is always located on the major axis It's always located on the major axis. It's always on the major axis Okay Yeah, so the major axis contains the foci it contains The vertices and of course the center So for the type 1 ellipse the foci is at plus minus a comma 0 for type 2 ellipse It is 0 comma plus minus B The equation of the directives is as you know directives These are the two directives So it's x equal to a by e and x equal to minus a by e and for the other one it's It's like this So it's y is equal to b by e and y equal to minus b by e just remember the change in the Expression for the eccentricity for the first one the eccentricity is given by e is under root 1 minus b square by a square Whereas for the second one it is given by e is equal to under root 1 minus a square by b square However irrespective of whichever case you are taking eccentricity will always be 1 minus the semi minor axis Square by semi major axis this square always so this is a universal formula which you can apply it to any type of ellipse right hope there is no question concern with respect to the Very basic fundamentals of ellipse which you have already learned in your class 11 Is that fine guys any question? Please type it in your chat box. Meanwhile, I can see eight people are attending this session Who are the other three please type in your names in the chat box? All right, so moving ahead to the next page length of the lattice victim for The case where you have the ellipse x square by a square plus y square by b square equal to 1 where your a is greater than b and For the other case where you have x square by a square plus y square by b square equal to 1 where your b is greater than a The length of the lattice rectum will change so for this one it is to B square by a square Sorry to b square by a and this is to a square by b Just remember the coordinates of the endpoint of the lattice rectums. It's actually lattice recta Okay, it's plus minus a comma plus minus b square by a and in this case it is Plus minus a square by b comma plus minus b These are the parametric representation for ellipse so you can write an ellipse as x equal to A cos phi y is equal to b sin phi for both these cases Focal radii focal radii means the distance of any point from the focus sp is equal to a minus Ex1 and from the other focus it is going to be a plus ex1 Similarly for the one where b is greater than a that is the second type of standard case. It is b minus e y1 Equal to b plus e y1 remember that in either of the two cases sp plus s dash p will be equal to the length of the major axis It will be the length of the major axis. That's actually another locus definition of an ellipse That's what has been written over here two a and two b This is between the foreside distance between the direct is is quite obvious tangents at the vertices is also quite obvious Okay So no question so far So I'll start today's session directly with the definition of a locus which you have already learned So the locus definition ellipse definition So ellipse is defined as the locus of a point which moves in a plane It's the locus of a point Which moves in a plane says that the ratio of its distance ratio of its distance from a fixed point From a fixed point That we call it as the focus to that from a fixed line that we call as the direct rakes is a constant That we call as the eccentricity that we call as the eccentricity written by e Where the value of e has to be somewhere between 0 and 1 Okay So on basis of this definition, I'm going to start up with a simple problem question Find the equation of an ellipse Find the equation of an ellipse of an ellipse whose focus is whose focus is minus 1 comma 1 eccentricity is half and direct rakes Equation is x minus y plus 3 equal to 0 So this question is based on your locus definition that we studied over here So please feel free to solve this and type in your response in the chat box So those who are attending the session request you all to type in your names in the chat box Guys, which exam do you have tomorrow? So guys here we'll use a definition that any point p is distance from the focus s Let's say I call this focus s as a point s Okay So s p by p m is always equal to e that is s p square is e square p m square Okay, so s p square is going to be the distance from the focus square of any point x comma y So that's going to be x plus 1 the whole square y minus 1 the whole square is equal to e square e square is 1 4th p m square is basically the distance of the point from The line x minus y plus 3 equal to 0 which is going to be this Okay, so if you just simplify this it's going to be 8 times x plus 1 whole square y minus 1 the whole square is equal to x square plus y square minus 2 x y minus 6 y plus 6 x and If you further open up over here, you get 8 times x square plus y square plus 2 x minus 2 y plus 2 Okay, so for the simplifying this you get 7 x square 7 y square You get 10 x you get plus 2 x y you get minus 10 y you get Yeah, there's a 9 missing out over here. So I just write a 9 The plus 7 Plus 7 equal to 0. So this is going to be your answer For this question. So guys, let me ask you another basic question before we move on find find center Coordinates of Center for psi for psi Let's say an equation of the directresses equation of the directresses for This ellipse Again, I'm sure you have learned about this as well This is the case of general form of an ellipse whose center is not at origin But however the major and the minor axis is are still parallel to the the x and the y axis So we had done a lot of problems last year based on this So just a quick recap of this through a problem find the center for psi and equation of the directresses For this ellipse, in fact, you also find out the length of the lattice rectum And please feel free to type in your response in the chat box Done any response for the lattice rectum length? Alright, so in the interest of time, I'll quickly solve this. So first of all, we take the x terms together We take the y terms together Okay, take 4 common it becomes x square minus 2x take 9 common It becomes y square minus 4 y Plus 4 equal to 0 which becomes 4 times x minus 1 the whole square minus 1 9 times y minus 2 square minus 4 Plus 4 equal to 0 So it's 4 times x minus 1 the whole square plus 9 times y minus 2 the whole square and We have a minus 4 minus 36. So 36 can be brought on here. So divide throughout with 36 You are going to get this equation for the ellipse correct now Since a this is your a square. This is your b square Since your a is greater than b Please recall the standard form where a is greater than b, right? So where is what is the length of the lattice rectum? First of all length of the lattice rectum for this case will be 2 b square by a So that's going to be 2 times b square which is 4 by a which is going to be 3 So length of the lattice rectum is 8 by 3 Okay Center center is you say x is 0 y is 0 that means the role of x now see guys here We have to do a role change The role of x is now being played by The role of x is now being played by x minus 1 and the role of y is now being played by Y minus 2 so when you say x equal to 0 and y equal to 0 it means x minus 1 is equal to 0 and y minus 2 equal to 0 Which implies x is 1 and y is 2 that means the center is at 1 comma 2 The center is at 1 comma 2 Okay Next is the 4 psi for before 4 psi you have to find the eccentricity eccentricity is 1 minus b square by a square in this case So that's under root of 1 minus 4 by 9 which is root 5 over 3 Okay 4 psi 4 psi for such cases x equal to plus minus ae and y is equal to 0 right So x equal to plus minus ae ae is this term and y is equal to 0 is this term which means it can be 1 plus root 5 and It can be 1 minus root 5 and y equal to 2 so the 2 4 psi possible in this case is The 2 4 psi possible in this case is 1 plus root 5 comma 2 and 1 minus root 5 comma 2 Okay What next we want we want the equation of the direct to sis Equation of the direct to sis is x equal to plus minus a by e So x is equal to plus minus a which is going to be 3 by e so it becomes 9 by root 5 So you can say x equal to 1 plus 9 by root 5 and x equal to 1 minus 9 by root 5 These are your direct to sis. These are your direct to sis Okay, so guys simple