Question

Give the standard equation of a sphere of radius \(r\), centered at \(\left(x_{0}, y_{0}, z_{0}\right)\)

Short answer

The standard equation of a sphere with center at \((x_{0}, y_{0}, z_{0})\) and radius \(r\) is \((x - x_{0})^{2} + (y - y_{0})^{2} + (z - z_{0})^{2} = r^{2}\).

Step-by-step solution

Recognise The Geometric Representation

For a sphere, the standard, general equation is given by \((x - x_{0})^{2} + (y - y_{0})^{2} + (z - z_{0})^{2} = r^{2}\), where \((x_{0}, y_{0}, z_{0})\) are the coordinates of the center of the sphere and \(r\) is the radius of the sphere.

Placing Values In The Equation

In the given problem, the center of sphere is \((x_{0}, y_{0}, z_{0})\) and its radius is \(r\). We simply need to replace these values into the general equation of the sphere.

Write Down The Final Equation

After placing the values, the standard equation becomes \((x - x_{0})^{2} + (y - y_{0})^{2} + (z - z_{0})^{2} = r^{2}\).