00:01
Okay, let's go ahead and try to step through this problem.
00:04
And we are given that we have a point p on an ellipse.
00:17
And we're going to let d be the distance.
00:24
And i'll draw a picture here in a minute or draw a graph of the ellipse from the center of the ellipse to the line tangent at point p and what we want to do is we want to prove that the distance from point p to one of the fosi times the distance from point p to the second foci times this d squared times the distance squared from the center of the ellipse to that line tangent at p is constant as p varies on the ellipse.
01:33
Okay, and so what we're going to do is we're going to get a little bit creative.
01:40
And so the first thing we're going to do is we're going to actually define the basically ellipse as x squared over a squared plus y squared over b squared and equal to one and so i'm going to go ahead and draw a picture which we're going to have to keep coming back to and so here we go let's just make this i'm going to try to make a bigger picture so that's negative a and a and this will be b and negative b.
02:12
So here that ellipse, whoops, not drawn, too great.
02:23
And so, and here is that point p.
02:29
And let's go ahead and label that point p.
02:33
Let's just label him x and y.
02:37
And then what we want is we know that we have this line segment that we're going to label d.
02:48
So the distance.
02:49
From the origin to that point.
02:51
And then what we're going to also do is if this is or these are the fosi, and so we're going to let that be fosi 1 and fosi 2.
03:05
And so if i actually draw line segments to make things easy, i'm going to label those as r2.
03:15
So this is going to be r2.
03:17
And this one is going to be, excuse me, r1.
03:25
So there we go.
03:27
And so what we want to do is to say that r1 times r2 times that distance squared is constant no matter where i put that point.
03:41
And so the first thing i want to do is to, we know that one, that for an ellipse, r1 plus r2 is equal to 2a.
03:58
And remember, that's because it's r1 and r2 are the distance from a point on the ellipse to each of the fosi.
04:06
So those, the addition of those two distances will actually equal 2a, whereas in a hyperbola, the difference of those two distances equals 2a.
04:16
Okay, and then this is going to be the line tangent.
04:24
So we're actually going to draw a tangent, whoops, if i can get that drawn, a tangent line, and we're going to extend them out.
04:36
So here's my tangent line.
04:38
Okay, so let's first on working on this r1 times r2 scenario.
04:44
And see if we can kind of work that out.
04:47
So we want to know what r1 times r2 is.
04:52
And we're going to actually be a little bit creative.
04:56
And i'm going to say i want to know what the twice r1 times r2 is.
05:03
And i'm going to let that be r1 plus r2 squared minus r1 squared minus r2 squared.
05:12
Okay.
05:14
And where i got that was, is because if i actually, so i'm going to do a side note here, if i actually square this and i get r1 squared plus 2, r1r2 plus r2 squared minus r2 squared minus r2 squared, you notice that that this 2 r1, this 2 r2 actually will equal, will actually equal this equation right here.
06:08
Because when i do that, i actually get 2r1r2 is equal to equal to um r1 squared um plus r2 squared minus r1 squared minus r2 squared and so um that will actually equal so when i do that these cancel so uh two two r one or two um and i don't want to do this so that will actually these two will um add up to zero, these two will add up to zero.
07:00
And so this side actually does equal to r1 r2, just kind of in a different way.
07:06
And so then that means what we have here is that r1 times r2 is one half.
07:24
Well, we also know that r1 plus r2 is 2a.
07:29
And so this actually becomes 1ā times 4a squared, minus r1 squared minus r2 squared.
07:39
Okay, and so now i'm going to kind of look at what r1 and r2 actually equal.
07:45
So that means i'm going to have to go back to, and we have a blank page, actually go back to our picture right here.
07:56
And i notice that r1, if i draw a right triangle here, r1 is the hypotenuse for this triangle right here, which is height is y, and its base length is actually going to be the x value plus c, because remember f1 is negative c, comma, zero, and f2 is c, c, 0 .0.
08:41
So this distance is x plus c.
08:44
So this is a hypotenuse.
08:46
And so r1 squared, r1 squared is actually the length of that hypotenuse.
08:58
And so that is going to be minus x plus c squared minus y squared.
09:06
And so now i'm going to go ahead and look at the second triangle, which is right here.
09:15
And i notice that once again, this is a right triangle where r2 is that hypotenuse of the triangle that is created from y and this distance right here.
09:26
And that distance is actually going to be x minus c, because it's going to be this x value here minus c.
09:37
And so now what i have, is this is minus x minus c squared minus the y squared.
09:48
And so this is this actually is r2 squared and this is r1 squared.
09:55
And so now i'm going to go ahead and foil things out and combine like terms and so forth.
10:00
So i get one half, four a squared.
10:04
This is minus x squared minus 2xc minus c squared, minus y squared minus y squared minus x squared plus 2xc minus c squared minus y squared.
10:23
And so you notice that the two xc's add up to zero...