00:01
The auxiliary circle of an ellipse is defined to be the circle with diameter the same as the major axis of the ellipse.
00:12
We want to determine the equation of the auxiliary circle for the ellipse 9x squared plus 25y squared equals 225.
00:22
That's part a.
00:23
For b, we would graph the ellipse 9x squared plus 25y squared equal to 125 along with its auxiliary circle.
00:37
Okay, so we have to determine the major axis of the ellipse.
00:45
We know already the center of the ellipse is 0, 0, so that will be the same center for the circle.
00:54
So we need to determine the major axis of the ellipse and for that we work in this equation so let me say here determine the major axis of the ellipse okay so to do that we need the equation of the ellipse in general is of the form 8x square over a square plus that is with the center of the ellipse in the origin which we know already looking at this expression because we have x square and y square there's no displacement of the x or y center so the center is at the origin and so the ellipse at the origin has the general form x square over a square plus y square over b square equal one so we got to find the equation that form so let's start with 9 is 9x square plus 25 y square equal to 125 that's the given equation and we know if 255 sorry 225 is 15 square so so the equation is 9x squared plus 25y squared equal 15 squared.
02:40
Now to have 1 on the right -hand side of the equation, we divide by 15 squared or equivalently by 225.
02:47
So 9x squared plus 25y squared divided by 15 squared is equal to 15 squared divided by 15 squared.
03:06
That is, now we separate these two terms into the denominator on the left -hand side of the equation to get 9x squared over 15 squared plus 25y squared over 15 squared is equal to 1.
03:28
But now to have in the numerator of this fraction x squared only, we write that fraction as x squared divided by 15 squared over 9.
03:42
That's just the same as this fraction written this way.
03:45
And we do the same in the other fraction here that is we write it as y square divided by the fraction 15 squared divided by 25 that's equal to 1 and now you see that the denominator of this fraction which is 15 square over 9 remember that 9 is 2 sorry 3 square so it's the same as this and the other one it's just the same as 15 squared, okay? 15 squared divided by 5 squared, which is 25.
04:32
And now we have a square in both numerator and denominator of this fraction, so we can put it like 15 thirds squared, because square in a fraction is equal to the fraction of 10 by square in the numerator and the denominator, plus y squared divided by and here we do the same 15 over 5 square let me clarify this over 5 square and that is equal to 1 so x squared divided by 15 over 3 is 5 so we get 5 square plus y square over 15 over 5 is 3 in that square equal to 1 this is the equation you are looking for so we see that the greatest fire at denominator is 5 square so the major axis is on the x axis so the major axis of the ellipse we know is exactly on the x -axis in this case because the center of the ellipse is 0, 0.
05:59
So the ellipse is like this.
06:02
So the major axis is on the x -axis.
06:10
And the length is 10 because we have the endpoints are 5, 0 and negative 5, 0.
06:19
So that is the shape, not a perfect graph at this moment of the ellipse is something like this or less let's say with this point here being 5 0 because when y equals 0 we the equation is x square over 5 square equal 1 that is x square equal 5 square so x is more or less 5 that is we have two points which are the intersection of the ellipse with the x -axis and we have two other points the intersection of the list with the y -axis that is putting x equals 0 we get y more or less 3 so we get this point is 0 3 and this one here is 0 negative 3 that's it and the center of the ellipse is 0 0 so the center of the ellipse is 0 0 so the center of the auxiliary circle is also 0 0 and the radius to let's see here the diameter of the auxiliary circle is the same as the major axis so be the diameter of the circle the major axis so the major axis of the ellipse is this one here and so the auxiliary circle will be circled with radius 5 and center the origin so so something like this...