Question

Find an equation of the sphere that has endpoints of a diameter at A and B

A(1,4,-2) B(-7,1,2)

Answer 1

The centre of the sphere is the midpoint of the diameter

$$x_C=\dfrac{x_A+x_B}{2}=\dfrac{1-7}{2}=-3$$

$$y_C=\dfrac{y_A+y_B}{2}=\dfrac{4+1}{2}=2.5$$

$$z_C=\dfrac{z_A+z_B}{2}=\dfrac{-2+2}{2}=0$$

The center of the sphere is $C=(-3, 2.5, 0).$

The length of the diameter is 

$$d=2r=\sqrt{(-7-1)^2+(1-4)^2+(2-(-2))^2}$$

$$=\sqrt{89}$$

So the radius of the sphere is $\sqrt{89}/2.$

The equation of the sphere is

$$(x+3)^2+(y-2.5)^2+z^2=22.25$$