# How do you write the equation of the circle with diameter that has endpoints at (–4, –1) and (–8, –9)?

Jun 23, 2018

${\left(x + 6\right)}^{2} + {\left(y + 5\right)}^{2} = 20$

#### Explanation:

To write the equation of a circle we need to know the center and the radius. Since we know two endpoints, we know the extrema of a diameter.

The idea is the following: we will find the center as the midpoint of the diameter, and the radius as half the length of the diameter.

To find the midpoint of a segmente with extrema $A = \left({x}_{A} , {y}_{A}\right)$ and $B = \left({x}_{B} , {y}_{B}\right)$, we have the following formula:

$M = \left(\setminus \frac{{x}_{A} + {x}_{B}}{2} , \setminus \frac{{y}_{A} + {y}_{B}}{2}\right)$

$C = \left(\setminus \frac{- 4 - 8}{2} , \setminus \frac{- 1 - 9}{2}\right) = \left(- 6 , - 5\right)$

As for the length, we have the following formula:

$d = \sqrt{{\left({x}_{A} - {x}_{B}\right)}^{2} + {\left({y}_{A} - {y}_{B}\right)}^{2}}$

$d = \sqrt{{\left(- 4 + 8\right)}^{2} + {\left(- 1 + 9\right)}^{2}} = \sqrt{16 + 64} = \sqrt{80} = 4 \sqrt{5}$

So, the diameter is $4 \sqrt{5}$ units long, which means that the radius is $2 \sqrt{5}$ units long.

Now we can write the equation:

in general, given the center $\left({x}_{0} , {y}_{0}\right)$ and the radius $r$, we have

${\left(x - {x}_{0}\right)}^{2} + {\left(y - {y}_{0}\right)}^{2} = {r}^{2}$

Which in this case becomes

${\left(x + 6\right)}^{2} + {\left(y + 5\right)}^{2} = 20$