Question

Find the equation of the surface.
The bottom hemisphere of a sphere centered at (4, -5, 0) with radius 10.

Answer 1

The equation of a spherical surface centred at (0,0,0) is
x^2+y^2+z^2=r^2
centred at (x0,y0,z0)
(x-x0)^2+(y-y0)^2+(z-z0)^2=r^2
substituting C(4,-5,0),
the equation of the spherical surface is
(x-4)^2+(y+5)^2+z^2=10^2=100

Assuming z-coordinate is positive=up, then
we have
(x-4)^2+(y+5)^2+z^2=100 (z<=0)
for the bottom hemisphere.