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

Answer 1

The major axis will be along the y-axis
with c = 4, 2a = 10
a = 5
by Pythagoras for this ellipse, b = 5

so ...

x^2/9 + y^2/25 = 1

Answer 2

Goci=foci, the plural of focus.