# How do you find the equation of a circle that passes through points (-8,-2)(1-,-11) and (-5,9)?

Jun 1, 2018

The equn. of a circle is :

$7 {x}^{2} + 7 {y}^{2} - 52 x - 10 y - 912 = 0$

#### Explanation:

Let the equation of the circle be

${x}^{2} + {y}^{2} + 2 g x + 2 f y + c = 0 ,$

Since the circle passes through $\left(- 8 , - 2\right) , \left(1 , - 11\right)$ and $\left(- 5 , 9\right)$

${\left(- 8\right)}^{2} + {\left(- 2\right)}^{2} - 16 g - 4 f + c = 0$
${1}^{2} + {\left(- 11\right)}^{2} + 2 g - 22 f + c = 0$
${\left(- 5\right)}^{2} + {9}^{2} - 10 g + 18 f + c = 0$

These equations simplify to

$- 16 g - 4 f + c + 68 = 0$ $- - - - - - - - - - - \left(1\right)$
$2 g - 22 f + c + 122 = 0$ $- - - - - - - - - - - \left(2\right)$
$- 10 g + 18 f + c + 106 = 0$ $- - - - - - - - - - \left(3\right)$

Subtracting $\left(1\right)$ from $\left(2\right)$
$18 g - 18 f + 54 = 0$ $- - - - - - - - - - - - \left(4\right)$

Subtracting $\left(1\right)$ from $\left(3\right)$
$6 g + 22 f + 38 = 0$ $- - - - - - - - - - - - - \left(5\right)$

Multiply $\left(5\right)$ by 3
$18 g + 66 f + 114 = 0$ $- - - - - - - - - - - - \left(6\right)$

Subtracting $\left(4\right)$ from $\left(6\right)$
$84 f + 60 = 0$

$\therefore f = - \frac{5}{7}$

Substituting the value of $f$ into $\left(4\right)$

$18 g + \frac{90}{7} + 54 = 0$
$18 g + \frac{468}{7} = 0$
$18 g = - \frac{468}{7}$

Divide both sides by 18
$g = - \frac{26}{7}$

Substituting the value of f and g into $\left(1\right)$
$- 16 \left(- \frac{26}{7}\right) - 4 \left(- \frac{5}{7}\right) + c + 68 = 0$
$\frac{416}{7} + \frac{20}{7} + c + 68 = 0$
$c + \frac{912}{7} = 0$
$c = - \frac{912}{7}$

Hence, the equation of the circle is

${x}^{2} + {y}^{2} + 2 \left(- \frac{26}{7}\right) x + 2 \left(- \frac{5}{7}\right) y - \frac{912}{7} = 0$
${x}^{2} + {y}^{2} - \frac{52}{7} x - \frac{10}{7} y - \frac{912}{7} = 0$

Multiply through by 7
$\therefore 7 {x}^{2} + 7 {y}^{2} - 52 x - 10 y - 912 = 0$

In the form
${\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}$

${\left(x - \frac{26}{7}\right)}^{2} + {\left(y - \frac{5}{7}\right)}^{2} = \frac{7085}{49}$