# How do you use the information provided to write the equation of each circle:Three points on the circle are (-15, -6), (-19,-4), and (-17, -6)?

Feb 6, 2017

Equation of only circle is ${x}^{2} + {y}^{2} + 32 x + 6 y + 255 = 0$

#### Explanation:

As there are three points on the circle are $\left(- 15 , - 6\right)$, $\left(- 19 , - 4\right)$ and $\left(- 17 , - 6\right)$, there is only one circle and the center of circle will lie on perpendicular bisector of any two sides.

Let us find the equation of perpendicular bisector of points joining $\left(- 15 , - 6\right)$ and $\left(- 19 , - 4\right)$.

As a point on a perpendicular bisector is equidistant from $\left(- 15 , - 6\right)$ and $\left(- 19 , - 4\right)$, its equation will be

${\left(x + 15\right)}^{2} + {\left(y + 6\right)}^{2} = {\left(x + 19\right)}^{2} + {\left(y + 4\right)}^{2}$

or ${x}^{2} + 30 x + 225 + {y}^{2} + 12 y + 36 = {x}^{2} + 38 x + 361 + {y}^{2} + 8 y + 16$

i.e. $30 x - 38 x + 12 y - 8 y = 377 - 261$

or $- 8 x + 4 y = 116$ or $2 x - y = - 29$ ....................(1)

Similarly equation of perpendicular bisector of points joining $\left(- 17 , - 6\right)$ and $\left(- 19 , - 4\right)$ is

${\left(x + 17\right)}^{2} + {\left(y + 6\right)}^{2} = {\left(x + 19\right)}^{2} + {\left(y + 4\right)}^{2}$

or ${x}^{2} + 34 x + 289 + {y}^{2} + 12 y + 36 = {x}^{2} + 38 x + 361 + {y}^{2} + 8 y + 16$

i.e. $34 x - 38 x + 12 y - 8 y = 377 - 325$

or $- 4 x + 4 y = 52$ or $x - y = - 13$ ....................(2)

Subtracting equation (2) from (1), we get

$x = - 16$ and hence putting this in (2), we get $y = - 3$

Hence center of the circle is $\left(- 16 , - 3\right)$ and square of the radius is square of the distance of center from say $\left(- 17 , - 6\right)$ i.e.

${\left(- 16 + 17\right)}^{2} + {\left(- 3 + 6\right)}^{2} = 1 + 9 = 10$

Hence equation of circle is

${\left(x + 16\right)}^{2} + {\left(y + 3\right)}^{2} = 10$

or ${x}^{2} + 32 x + 256 + {y}^{2} + 6 y + 9 = 10$

or ${x}^{2} + {y}^{2} + 32 x + 6 y + 255 = 0$
graph{x^2+y^2+32x+6y+255=0 [-20.645, -10.645, -6.88, -1.88]}