What is the equation of the ellipse with foci at (0, -4) and (0, 4) and the sum of its focal radii being 10?

Question

Please help! I can not figure this problem out!

Answer 1

The foci of this ellipse are on the y-axis, so it's vertically oriented. "a" will be the length of the major, vertical axis. "b" will be the length of the minor, horizontal axis. "c" is the focal length, which in this case is 4. The center of this ellipse is the origin (0,0).

 

The sum of the focal radii is the distance from one focus to any point on the ellipse plus the distance from that same point to the other focus. This makes a sort of V from one focus to the edge back to the other focus. If you consider the point at the end of the major axis, you'll find that this distance is (c+a) + (a-c) = 2a. (If you draw the ellipse and trace these distances, hopefully you'll see where these expressions come from.

 

Since this sum is 10 = 2a, then a=5 is the length of the major axis.

 

For an ellipse, a b and c are related by

a² - b² = c²

and you can solve for b now that you know a and c.

 

Your equation will then be of the form:

 

y²/a² + x²/b² = 1

 

Substitute in your values for a and b, and you're done!

 

Hope this helps, please comment if you have further questions!