What is the equation of the ellipse with foci at (0,-3) and (0,3) and the sum of focal radii being 10?

I tried it two different ways and am not sure if either is correct:

First way:

2a=10 so a=5 (sum of focal radii)

Center is (0,0) halfway between foci

c=3

b^2=a^2-c^2 or b^2 = 16

giving me the equation x^2/16 + y^2/25 = 1

Then I tried it like the textbook by using the distance formula:

squareroot of [x² + (y+3)²] + squareroot of [x² + (y-3)²]=10

When I tried to square both sides, I ended up getting it simplified to x²+y²= 41

Obviously totally different answers—not sure which is correct and why the other is incorrect.

Thank you for your assistance.

I tried it two different ways and am not sure if either is correct:

First way:

2a=10 so a=5 (sum of focal radii)

Center is (0,0) halfway between foci

c=3

b^2=a^2-c^2 or b^2 = 16

giving me the equation x^2/16 + y^2/25 = 1

Then I tried it like the textbook by using the distance formula:

squareroot of [x² + (y+3)²] + squareroot of [x² + (y-3)²]=10

When I tried to square both sides, I ended up getting it simplified to x²+y²= 41

Obviously totally different answers—not sure which is correct and why the other is incorrect.

Thank you for your assistance.

## Comments

^{2}=a^{2}+b^{2}-2ab^{2}= 10^{2}+ (squareroot of [x² + (y-3)²])^{2}-2*10*(squareroot of [x² + (y-3)²]) = 100 + x² + (y-3)² -20*squareroot of [x² + (y-3)²]