If you're having trouble with ellipses,
. . perhaps you don't understand the basic concepts.
But I must admit: this problem can be somewhat tricky.
Find an equation for the ellipse centered at the origin
with one focus (0,-3) and the sum of the focal radii equal to 10. Sketch.
An ellipse centered at the origin has the equation: .
where is the "x-radius" and is the "y-radius".
is the "focal distance", distance from the center to a focus.
. . Formula: .
Since one focus is (0,-3), the other is (0,3) . . . and
We also know that the ellipse is "horizontal": .
* * * P
* | *
* | * * *
* * * *
* | *
* - - * - - - + - - - * - - *
F2 | F1 a
The "focal radii" are the distances
. . We are told that: .
Here's a lesser-used fact: .
. . The sum of the focal radii is twice the semimajor axis.
So we have:. .
From the formula, we have: .
Therefore, the equation is: .