Question

Find the equation of a sphere with center (2, 5,-7) and containing the origin.

Short answer

The required equation is${(x-2)}^2+{(y-5)}^2+{(z+7)}^2=78$

Step-by-step solution

Given information

A sphere with center (2, 5, -7) and containing the origin.

Finding the radius of sphere

The radius will be distance between the origin (through which the sphere passes) and center of sphere.

Radius is:

$\sqrt{{(0-2)}^2+{(0-5)}^2+{(0+7)}^2}$

Simplifying:

$\sqrt{78}$

Finding the equation

The equation of sphere is:

${(x-2)}^2+{(y-5)}^2+{(z+7)}^2={\left(\sqrt{78}\right)}^2$

Simplify:

${(x-2)}^2+{(y-5)}^2+{(z+7)}^2=78$