Question

Find the centre of a circle passing through points $(6,-6),(3,-7)$ and $(3,3)$

Answer 1

Find the Centre of Circle:

Let Let $O$ be the centre of circle whose co-ordinate is $(x,y)$ .

The circle passing through points $(6,-6),(3,-7)$ and $(3,3)$

Let $A=(6,-6),B=(3,-7)$and $C=(3,3)$

If $O$ is the centre, then $OA=0B=OC$ (radii are equal)

From $OB=OC$ we get

$$\begin{gathered} {\left(x-3\right)}^2+{\left(y+7\right)}^2={\left(x-3\right)}^2+{\left(y-3\right)}^2 \\ \Rightarrow {\left(y+7\right)}^2={\left(y-3\right)}^2 \\ \Rightarrow y^2+14y+49=y^2-6y+9 \\ \Rightarrow 20y=-40 \\ \Rightarrow y=-2 \end{gathered}$$

From $OA=OB$ we get

$$\begin{gathered} {\left(x-6\right)}^2+{\left(y+6\right)}^2={\left(x-3\right)}^2+{\left(y+7\right)}^2 \\ \Rightarrow x^2-12x+36+y^2+12y+36=x^2-6x+9+y^2+14y+49 \\ \Rightarrow -6x=2y-14 \\ \Rightarrow 6x+2y=14 \end{gathered}$$

$\therefore x=3[\because y=-2]$

Hence, the centre of the circle located at point $(3,-2).$