SOLUTION: Find equation for the ellipse that shares a vertex and focus with the parabola x^2 + y = 100 and has its other focus at the origin? I know what the shape of the two figures toge

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Find equation for the ellipse that shares a vertex and focus with the parabola x^2 + y = 100 and has its other focus at the origin? I know what the shape of the two figures toge      Log On


   

Question 451902: Find equation for the ellipse that shares a vertex and focus with the parabola x^2 + y = 100 and has its other focus at the origin?
I know what the shape of the two figures together would look like, and i found that 2C is 100, since that is the distance between foci. From that I determined that 2b - c = 100. What I don't get is how to find a and b. Thanks very much!
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
You say that the ellipse shares a vertex (the point (0, 100)) and a focus with the parabola x%5E2+%2B+y+=+100, or y+=+-x%5E2+%2B+100, or 4%2A%281%2F4%29%2A%28y+-+100%29+=+-%28x-0%29%5E2. Hence for the parabola, the distance from the vertex to the focus is 1/4. Also, one focus of the ellipse must be (0, 399/4). It was given that the other focus of the ellipse is (0,0) (the origin). The center of the ellipse is then the point (0, 399/8). So
c+=+399%2F4+-+399%2F8+=+399%2F8, and a+=+100+-+399%2F8+=+401%2F8.
Then b%5E2+=+a%5E2+-+c%5E2+=+25
Then the standard equation is
x%5E2%2F%28401%2F8%29%5E2+%2B+%28y+-+399%2F8%29%5E2%2F25+=+1, or
%2864x%5E2%29%2F160801+%2B+%28y-399%2F8%29%5E2%2F25+=+1.