# Question #cfe32

Apr 4, 2017

$\textcolor{red}{7 {x}^{2} + 7 {y}^{2} - 67 x + 53 y + 216 = 0}$

#### Explanation:

The general equation of a circle is given as ${x}^{2} + {y}^{2} + 2 g x + 2 f y + c = 0$ where $g , f , c$ are constants .

Substituting the given three points one-by-one in the above equation,

1. $\left(6 , - 6\right)$
$36 + 36 + 12 g - 12 f + c = 0$
$\implies 72 + 12 g - 12 f + c = 0$

2.$\left(3 , - 2\right)$
$9 + 4 + 6 g - 4 f + c = 0$
$\implies 13 + 6 g - 4 f + c = 0$

3.$\left(7 , - 5\right)$
$49 + 25 + 14 g - 10 f + c = 0$
$\implies 74 + 14 g - 10 f + c = 0$

subtracting 2. from 3.;
$74 + 14 g - 10 f + c - \left(13 + 6 g - 4 f + c\right) = 0$
$\implies 61 + 8 g - 6 f = 0$ -------------------------( 4. )

subtracting 1. from 3.;
$74 + 14 g - 10 f + c - \left(72 + 12 g - 12 f + c\right) = 0$
$\implies 2 + 2 g + 2 f = 0$
$\implies 1 + g + f = 0$ ----------------------( 5. )

from 5., $g = - 1 - f$ ---------------------( 6. )

substituting this value of $g$ in 4.;
$61 + 8 \left(- 1 - f\right) - 6 f = 0$
$\implies 61 - 8 - 8 f - 6 f = 0$
$\implies 53 = 14 f$
$\implies \textcolor{red}{f = \frac{53}{14}}$

substituting this value of $f$ in 6.;
$g = - 1 - \frac{53}{14} = \frac{- 14 - 53}{14}$

$\implies \textcolor{red}{g = - \frac{67}{14}}$

substituting these values of $f$ and $g$ in any of the equations 1., 2., 3., to obtain the value of c.

Let's use 2.
$13 - 6 \cdot \frac{67}{14} - 4 \cdot \frac{53}{14} + c = 0$
$\implies - \frac{216}{7} + c = 0$
$\implies \textcolor{red}{c = \frac{216}{7}}$

substituting these values of $g , f , c$ in the general equation of a circle [${x}^{2} + {y}^{2} + 2 g x + 2 f y + c = 0$]

${x}^{2} + {y}^{2} - \frac{67}{7} x + \frac{53}{7} y + \frac{216}{7} = 0$

$\implies \textcolor{red}{7 {x}^{2} + 7 {y}^{2} - 67 x + 53 y + 216 = 0}$

is the required equation of the circle.