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

Feb 19, 2015

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

$\left(9 , 4 , - 6\right)$ relative to a center $\left(6 , 7 , 2\right)$
is equivalent to $\left(3 , - 3 , - 8\right)$ relative to $\left(0 , 0 , 0\right)$

The equation of a sphere through $\left(3 , - 3 , - 8\right)$ with center $\left(0 , 0 , 0\right)$ is

${X}^{2} + {Y}^{2} + {Z}^{2} = {\left(3\right)}^{2} + {\left(- 3\right)}^{2} + {\left(- 8\right)}^{2}$
or
${X}^{2} + {Y}^{2} + {Z}^{2} = 82$

For $\left(x , y , z\right) = \left(6 , 7 , 2\right)$ to be equivalent to $\left(X , Y , Z\right) = \left(0 , 0 , 0\right)$
$X = x - 6$
$Y = y - 7$
$Z = z - 2$

So the desired equations is:
${\left(x - 6\right)}^{2} + {\left(y - 7\right)}^{2} + {\left(z - 2\right)}^{2} = 82$