 Now we'll move on to the next concept which is the concept of equation of a chord Equation of a chord joining points Phi and theta And I say points phi and theta. They are called the eccentric angles. These are called the eccentric angles By the way guys all of you are familiar what are these eccentric angles if not, let me quickly tell you that so let's say this is a this is a ellipse Ascentic angle is basically when you draw a circle when you draw a circle this angle that You make from the line connecting the origin to any point on it. This angle is called the eccentric angle, okay? And when you drop this down on the ellipse then this point This point will have coordinates a cos theta comma b sin theta Please note that it is not the line connecting the center to any point It is first it goes to the circle and from there you drop down So that point eccentric angle will be theta. So theta is not the angle between the line connecting the origin directly with the ellipse so first it goes and hits this circle which we call as the Auxiliary circle this is called the auxiliary circle and from the auxiliary circle Let's say it hits the auxiliary circle at B Then from B, sorry at a and from a we drop a perpendicular down to hit the Ellipse at P. So P will be a cos theta comma B sin theta Okay Now my question here was if you have an ellipse if you have an ellipse if you have an ellipse like this What's the equation of the chord connecting any two points? Having the eccentric angle phi and theta Okay So what's the equation of this chord? Let me name it as AB So equation of the chord AB It's very simple. You know two points Correct when you know two points you can always find out the slope of the line connecting those two points And you can use the slope point form So guys, I will just save your time in this and I'll quickly discuss with you the result I'm sure you can always Find this out by using the slope point form of the equation of a line So, please note down the equation of the chord Connecting to eccentric points theta and phi. So this is the equation of the chord, right? So remember when your theta tends to phi or phi tends to theta when theta tends to phi or phi tends to theta It becomes x by a cos theta or x by a cos phi plus y by B sin phi is Equal to 1 so this is actually the equation of the tangent or the parametric equation of the tangent drawn at phi Anyways, we are going to talk about it separately in terms of tangents Why I asked you why I gave you this equation of the chord is because there's a very important question that comes associated with the equation of a chord The question goes like this if a focal chord if a focal chord Joining our two eccentric points. Let's say phi 1 and phi 2 two eccentric points phi 1 and phi 2 Passes through passes through a e comma 0 Okay, then prove that tan of phi 1 by 2 into tan of phi 2 by 2 is e minus 1 by e plus 1 and if it passes through minus e comma 0 then prove that tan phi 1 by 2 tan phi 2 by 2 is equal to e minus e Plus 1 by e minus 1 any one of them if you can prove that's fine. No need to prove both of them Please type done in your chat box if you're done with it so that I can start the discussion with it All right, so it should not be of much problem to you Let's say I want to prove the first one So I know the equation of the chord connecting two eccentric points phi 1 and phi 2 is x by a cos Phi 1 plus phi 2 by 2 plus y by b sin Phi 1 plus phi 2 by 2 equal to cos Phi 1 minus phi 2 by 2 Right now, it's also given in the question that it passes through a e comma 0 So substitute x as a and y is 0. So you get e cos Phi 1 by phi 2 by 2 is equal to cos Phi 1 minus phi 2 by 2 So e by 1 is equal to cos Phi 1 minus phi 2 by 2 by cos phi 1 plus phi 2 by 2 let's apply component oh and dividend oh that means let's do e plus 1 by sorry e minus 1 by e minus 1 by e plus 1 it means you have to Subtract the denominator from here. That is phi 1 plus phi 2 by 2 and In the denominator you have to add them both yes, so We will get e minus 1 by e plus 1 as this term will actually become Sign this plus this by 2 which is phi 1 by 2 into sine phi 2 by 2 And in the denominator you'll get to cos phi 1 by 2 cos phi 2 by 2 So 2 and 2 gets cancelled leaving you with tan phi 1 by 2 Into tan phi 2 by 2. Okay Great. So you can similarly prove the second one as well. You can similarly prove the second one as well Right. Hope it's clear. No doubt with respect to it If you have any doubt, please feel free to type it in the chat box All right Next can say that we are going to talk about is position of a point with respect to an ellipse position of a point With respect to an ellipse So let's say I have an ellipse like this now a point x1 y1 a Point x1 comma y1 could be at position a B or c that means it can be within a within the ellipse That means that position a on the ellipse that is position B or outside the ellipse that is position C Okay So when we say x square by a square plus y square by b square equal to 1 Please remember this this expression is actually called the expression s For an ellipse just like we had it in case of a circle and parabola Similarly, when you substitute a point in place of x1 that is called s1 the normal notation that we use so This is called the power of the point with respect to an ellipse So if the power of the point is negative that means the point is located position B And if this is positive then we say the point is located at C Okay, nothing very great about this concept. So we can move on next is a Intersection of a line in an ellipse Intersection you realize more or less the same concepts are repeated Intersection of a line and an ellipse So let's say we have a line and an ellipse line is basically the y equal to mx plus c form And ellipse is let's say our standard case Okay x square by a square plus y square equal to y square by b square equal to 1 Now a line in an ellipse a line in an ellipse can interact with each other in various ways it can either cut or It can touch or it cannot even touch Okay, so let's call this situation as situation a b and c Okay, so when you think the situation a will arise the situation a will arise when We when you directly solve these two when you simultaneously solve these two equations equation number one and two Right, you will get two real and distinct roots Right so when you substitute y is equal to mx plus c you get something like this And when you simplify this I'm directly going to write the quadratic that you are going to get you're getting going to get a square m square Plus b square x square plus 2 mc a square x Plus c square a square minus a square b square equal to 0 When you simplify this This is the quadratic that you're going to get in x So for it to have a real and distinct roots its discriminant should be greater than 0 That is your b square that is 4 m square c square a to the power 4 should be greater than 4 a c 4 a c Okay, I Think you can directly drop the factor of a square and 4 so 4 and a square will be dropped So you get m square a Is square m square c square is greater than a square m square c square minus a square m square b square Okay, plus b square c square minus b to the power 4 so these two will be cancelled off that means b to the power 4 B to the power 4 plus a square m square b square is Is a greater than b square c square? Okay, so drop the factor of b square again, so it becomes c square is less than less than a Square m square plus b square All right, so this is the condition for A to happen that means the line y equal to x is cutting the ellipse So without wasting much time we can say when c square is equal to a square m square the line will be touching the ellipse This is your number a this is your situation number b So it is touching the ellipse Touches the ellipse or it is a tangent to the ellipse Okay, and when your c square is c square is greater than a square m square plus b square it is going to Not touch the ellipse at all Please note that these condition are very much specific to the standard case of an ellipse So as the ellipse changes these conditions will also undergo a change of All these conditions situation number b is importance important to us which we call as the condition of tangency Which we call as the condition of tangency for any line y equal to mx plus c to touch x square by a square plus y square by b square equal to one So we'll talk a little bit more about it So condition of tangency is c square is equal to a square m square plus b square which actually implies c can be Plus minus under root a square m square plus b square In other words, you can also write the equation of a tangent You can also write the equation of a tangent as Y equal to mx plus c in place of c you can actually replace this So that means if you're given the slope and the equation of the ellipse you can always draw You can always write down the equation of the two tangents as follows So this form is also called the slope form of the equation of the tangent This one is called the slope form of the equation of the tangent. Okay, and It's very important to also note the point of contact the point of contact where these Tangents will touch the ellipse the point of contacts can be remembered by a very simple formula a square m plus minus a square m by c Plus minus b square by c plus minus b square by c Right where you see is actually this expression c is actually under root a square m square plus b square So this is the point of contact. Please remember this. It'll just save time for you. All right Of course, we'll be talking about other form of ellipses other form of tangents as well But meanwhile, I would like to take up a question on this. I would like to take The question is if x cos alpha Plus y sin alpha equal to p Touches the standard form of an ellipse Touches the standard form of ellipse and prove that a square cos square alpha Plus b square sin square alpha is equal to p square Please do this question and type done on your chat box once you're done. It's a super simple question. All right, so done So it's quite simple You first write it in the form of y equal to mx plus c so it becomes y is equal to mx So it becomes minus x cot alpha Plus p cosik alpha Okay So we can use the condition for tangency that is c square is equal to a square m square plus b square so c square which is going to be This is equal to a square m square which is going to be cot square alpha plus b square and You multiply throughout with sine square that's going to give you a square cos square alpha plus b square sine square alpha Right hence proved. Okay. Very simple question to start with