# A circle passes A(-4,-1) and B(2,1). How do you find the center and radius of this circle if the center of the circle is on y-axis?

Mar 28, 2018

graph{x^2 +(y+3)^2=20 [-10, 10, -5, 5]}
Given that the center of the circle is on y-axis ($x = 0$), let us consider the coordinates of its center $\left(O\right)$ be $\left(0 , a\right)$

So

$O {A}^{2} = O {B}^{2} = {R}^{2}$,Where R is the radius of the circle.

${4}^{2} + {\left(a + 1\right)}^{2} = {2}^{2} + {\left(a - 1\right)}^{2} = {R}^{2}$

$\implies {\left(a + 1\right)}^{2} - {\left(a - 1\right)}^{2} = {2}^{2} - {4}^{2}$

$\implies 4 a = - 12$

$\implies a = - 3$

Hence coordinates of the center (O) is (-3,0).

And radius $R = \sqrt{{2}^{2} + {\left(a - 1\right)}^{2}}$

$\implies R = \sqrt{{2}^{2} + {\left(- 3 - 1\right)}^{2}} = 2 \sqrt{5}$