Question

Use the midpoint formula to find the equations of the sphere in which the segment with endpoints (3, −2, 6) and (5, 6, 4) is a diameter.

Short answer

The required equation is${(x-4)}^2+{(y-2)}^2+{(z-5)}^2=18$

Step-by-step solution

Given information

A sphere in which the segment with endpoints (3, −2, 6) and (5, 6, 4) is a diameter.

Finding the radius and center

The radius is half of diameter which is half of the distance between the two endpoints given.

Radius is:

$\frac{1}{2}\sqrt{{(5-3)}^2+{(6+2)}^2+{(4-6)}^2}$

Solving:

Radius is $3\sqrt{2}$

The center, as per mid point formula, is:

$\left(\frac{5+3}{2},\frac{6-2}{2},\frac{4+6}{2}\right)$

Simplify:

$\left(4,2,5\right)$

Writing the equation

The equation of sphere is:

${(x-4)}^2+{(y-2)}^2+{(z-5)}^2={\left(3\sqrt{2}\right)}^2$

Simplify:

${(x-4)}^2+{(y-2)}^2+{(z-5)}^2=18$