Question

Use mid point formula to find the equation of the sphere in which the segment with endpoints (6, -1, 4) and (-3, 3, 1) is a diameter.

Short answer

The required equation is${\left(x-\frac{3}{2}\right)}^2+{(y-1)}^2+{\left(z-\frac{5}{2}\right)}^2=\frac{53}{2}$

Step-by-step solution

Given information

A sphere in which the segment with endpoints (6, -1, 4) and (-3, 3, 1) is a diameter.

Finding the center and radius

The center, as per mid point formula, is:

$\left(\frac{6-3}{2},\frac{-1+3}{2},\frac{4+1}{2}\right)$

Simplify:

$\left(\frac{3}{2},1,\frac{5}{2}\right)$

The radius will be half of diameter.

The radius is:

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

Simplify:

Radius =$\frac{\sqrt{106}}{2}$

Finding the equation

The equation of sphere may be written as:

${\left(x-\frac{3}{2}\right)}^2+{(y-1)}^2+{\left(z-\frac{5}{2}\right)}^2={\left(\frac{\sqrt{106}}{2}\right)}^2$

Simplify:

${\left(x-\frac{3}{2}\right)}^2+{(y-1)}^2+{\left(z-\frac{5}{2}\right)}^2=\frac{53}{2}$