How do you find an equation of the sphere that passes through the point (9, 4,-6) and has a center (6,7,2)?

1 Answer
Feb 19, 2015

Find the equation for the equivalent sphere centered at (0,0,0) and then shift the co-ordinates.

#(9,4,-6)# relative to a center #(6,7,2)#
is equivalent to #(3,-3,-8)# relative to #(0,0,0)#

The equation of a sphere through #(3,-3,-8)# with center #(0,0,0)# is

#X^2 + Y^2 + Z^2 = (3)^2 + (-3)^2 + (-8)^2#
or
#X^2 + Y^2 + Z^2 = 82#

For #(x,y,z) = (6,7,2)# to be equivalent to #(X,Y,Z) = (0,0,0)#
#X = x-6#
#Y = y-7#
#Z = z-2#

So the desired equations is:
#(x-6)^2 + (y-7)^2 + (z-2)^2 = 82#