# A circle has a diameter with endpoints (-10, -6) and (-2, -4). What is the equation of the circle?

Feb 9, 2016

${\left(x + 6\right)}^{2} + {\left(y + 5\right)}^{2} = 17$

#### Explanation:

A line segment with end points $\left(- 10 , - 6\right)$ and $\left(- 2 , - 4\right)$

• has a mid point at $\left(- 6 , - 5\right)$
• has a length of $\sqrt{{\left(- 10 - \left(- 2\right)\right)}^{2} + \left(- 6 - {\left(- 4\right)}^{2}\right)} = \sqrt{{8}^{2} + {2}^{2}} = \sqrt{68} = 2 \sqrt{17}$

If the line segment represents a diameter for a circle:
the circle must:

• have its center at $\left(\textcolor{red}{- 6} , \textcolor{b l u e}{- 5}\right)$
• have a radius of $\frac{2 \sqrt{17}}{2} = \textcolor{g r e e n}{\sqrt{17}}$

The general equation of circle with center at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$ and radius $\textcolor{g r e e n}{r}$ is
$\textcolor{w h i t e}{\text{XXX}} {\left(x - \textcolor{red}{a}\right)}^{2} + {\left(y - \textcolor{b l u e}{b}\right)}^{2} = {\textcolor{g r e e n}{r}}^{2}$

Substituting:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a = - 6}$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{b = - 5}$
color(white)("XXX")color(green)(r=sqrt(17)

We get:
$\textcolor{w h i t e}{\text{XXX}} {\left(x - \left(\textcolor{red}{- 6}\right)\right)}^{2} + {\left(y - \left(\textcolor{b l u e}{- 5}\right)\right)}^{2} = {\left(\textcolor{g r e e n}{\sqrt{17}}\right)}^{2}$
or
$\textcolor{w h i t e}{\text{XXX}} {\left(x + 6\right)}^{2} + \left(y + 5\right) = 17$

The graph below of this equation might help to verify that it does seem to match the given end point conditions.
graph{(x+6)^2+(y+5)^2=17 [-14.16, 5.84, -9.76, 0.235]}