Question

Points A$(-1,y)$ and B$(5,7)$ lie on a circle with centre O$(2,-3y)$. Find the values of y. Hence, find the radius of the circle.

Answer 1

Since O is the centre of the circle and A,B are points on its circumference.

$\therefore OA=OB=Radius$

$\Rightarrow OA=OB$

$\Rightarrow \sqrt{{(2+1)}^2+{(-3y-y)}^2}=\sqrt{{(2-5)}^2+{(-3y-7)}^2}$

$\Rightarrow 9+{16y}^2=9+{(3y+7)}^2$
$\Rightarrow {16y}^2={9y}^2+42y+49$
$\Rightarrow {7y}^2-42y-49=0$
$\Rightarrow y^2-6y-7=0$
$\Rightarrow (y-7)(y+1)=0\Rightarrow y=-1,7$

When y=-1: The coordinates of O,A and B are O(2,3), A(-1,-1) and B(5,7) respectively.

$\therefore Radius=OA=\sqrt{{(2+1)}^2+{(3+1)}^2}=5$

When y=7: The coordinates O,A and B are O(2,-21), A(-1,7) and B(5,7) respectively.

$\therefore Radius=OA=\sqrt{{(2+1)}^2+{(-21-7)}^2}=\sqrt{9+784}=\sqrt{793}$