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Equations of ellipse

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
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.

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Francisco E. | Francisco; Civil Engineering, Math., Science, Spanish, Computers.Francisco; Civil Engineering, Math., Sci...
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first you can not square both sides as easy as that ans second The first equation is 
(x2/25) + (Y2/16) = 1
16x2 + 25y2 = 400