# How do you find the equation of a circle whose diameter endpoints of (5,8) and (5,-4)?

Nov 26, 2016

#### Explanation:

The standard form for the equation of a circle is:

${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$

where $x \mathmr{and} y$ are any point, $\left(x , y\right)$, on the circle, $\left(h , k\right)$ is the center, and r is the radius.

The center of the circle is halfway between the two endpoints, from $\left(5 , 8\right)$ to $\left(5 , - 4\right)$:

$\Delta x = \left(5 - 5\right) = 0$
$\Delta y = \left(- 4 - 8\right) = - 12$

This means that the center is $- 6$ in the vertical direction from the starting point, $\left(5 , 8\right)$, which brings us to the point $\left(5 , 2\right)$.

This, also, tells us that the radius is 6.

We have have all of the information that we need to substitute into the standard form:

${\left(x - 5\right)}^{2} + {\left(y - 2\right)}^{2} = {6}^{2}$