Question

Solve
$(5,-8),(2,-9)$ and $(2,1)$

Answer 1

Let the given points be $A(5,-8)$,$B(2,1)$ $C(2,-9)$ and centre of the circle be $O(x,y)$

Then $OA=OB$ (radii of the same circle )

$\Rightarrow {\left(OA\right)}^2={\left(OB\right)}^2$

$\Rightarrow {\left(x-5\right)}^2+{\left(y+8\right)}^2={\left(x-2\right)}^2+{\left(y-1\right)}^2$

$\Rightarrow 25-10x+64+16y=4-4x+1-2y$

$\Rightarrow x-3y-14=0$ --(i)

Also, $OB=OC$ (radii of the same circle)

$\Rightarrow {\left(OB\right)}^2={\left(OC\right)}^2$

$\Rightarrow {\left(x-2\right)}^2+{\left(y-1\right)}^2={\left(x-2\right)}^2+{\left(y+9\right)}^2$

$\Rightarrow 5-2y=85+18y$

$\Rightarrow 20y=-80$

$\Rightarrow y=-4$

putting $y=-4$ in (i)

$\Rightarrow x-3\times -4-14=0$

$\Rightarrow x=2$

Hence the centre of the circle is $(2,-4)$