Question

Given a sphere in which the segment with endpoints (1, 5, -2) and (3, 9, -1) is a diameter.

Short answer

The required equation is:

${(x-2)}^2+{(y-7)}^2+{\left(z+\frac{3}{2}\right)}^2=\frac{21}{4}$

Step-by-step solution

Given information

A sphere with the segment with endpoints (1, 5, -2) and (3, 9, -1) is a diameter.

Finding the center and radius

The center may be found using the mid point formula.

The center is:

$\left(\frac{1+3}{2},\frac{5+9}{2},\frac{-2-1}{2}\right)$

Simplify:

$\left(2,7,\frac{-3}{2}\right)$

The radius will be half of the diameter. Using the distance formula to find the radius:

$\frac{1}{2}\sqrt{{(3-1)}^2+{(9-5)}^2+(-1-{(-2))}^2}$

Simplify:

$\frac{\sqrt{21}}{2}$

Writing the equation

The equation of sphere may be written as:

${(x-2)}^2+{(y-7)}^2+{\left(z-\left(-\frac{3}{2}\right)\right)}^2=\left(\frac{\sqrt{21}}{2}\right)$

Simplify:

${(x-2)}^2+{(y-7)}^2+{\left(z+\frac{3}{2}\right)}^2=\frac{21}{4}$