# A circle has a center that falls on the line y = 2/3x +1  and passes through (5 ,2 ) and (3 ,2 ). What is the equation of the circle?

Aug 21, 2016

${\left(x - 4\right)}^{2}$+ ${\left(y - \frac{11}{3}\right)}^{2}$ =$\frac{34}{9}$

#### Explanation:

The general equation of a circle is
${\left(x - a\right)}^{2}$ + ${\left(y - b\right)}^{2}$=${r}^{2}$
Where (a,b) is the centre of the circle and r is the radius.

So (a,b) is on the line y=$\frac{2}{3}$ x +1
Substituting b=$\frac{2}{3}$a+1. Equation 1

(5,2) is on the circle so
${\left(5 - a\right)}^{2}$ +${\left(2 - b\right)}^{2}$=${r}^{2}$. Equation 2

(3,2) is on the circle so
${\left(3 - a\right)}^{2}$+${\left(2 - b\right)}^{2}$ =${r}^{2}$ Equation 3

Subtract equation 3 from equation 2 gives
${\left(5 - a\right)}^{2}$ -${\left(3 - a\right)}^{2}$ =0

Multiply out and simplify gives a= 4

Substitute in equation 1 gives b= $\frac{11}{3}$

Substitute in equation 2 gives ${r}^{2}$ =$\frac{34}{9}$

Put all values into equation 3 as a check.

Yes it is correct