# How do you write the equation of the circle whose centre is at (-5, 3) and which passes through the point (-4, -5)?

Nov 3, 2017

${\left(x + 5\right)}^{2} + {\left(y - 3\right)}^{2} = 65$

#### Explanation:

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

We are given
$\textcolor{w h i t e}{\text{XXX}} a = - 5$ and
$\textcolor{w h i t e}{\text{XXX}} b = + 3$

so we only need to find the radius.
The radius is the distance from the center to any point on the circumference.
Given the center, $\left(- 5 , 3\right)$, and a point on the circumference, $\left(- 4 , - 5\right)$ we can evaluate the radius using the Pythagorean Theorem
$\textcolor{w h i t e}{\text{XXX}} {r}^{2} = {\left(- 5 - \left(- 4\right)\right)}^{2} + {\left(3 - \left(- 5\right)\right)}^{2}$

$\textcolor{w h i t e}{\text{XXX=}} = {\left(- 1\right)}^{2} + {8}^{2}$

$\textcolor{w h i t e}{\text{XXX=}} = 65$

Therefore the equation of the circle is
$\textcolor{w h i t e}{\text{XXX}} {\left(x - \left(- 5\right)\right)}^{2} + {\left(y - 3\right)}^{2} = 65$
or, simplifying the first term
$\textcolor{w h i t e}{\text{XXX}} {\left(x + 5\right)}^{2} + {\left(y - 3\right)}^{2} = 65$