Question

Points A(−1, y) and B(5, 7) lie on a circle with centre O(2, −3y). Find the values of y. [CBSE 2014]

Answer 1

The given points are A(−1, y), B(5, 7) and O(2, −3y).
Here, AO and BO are the radii of the circle. So
$$\begin{gathered} AO=BO\Rightarrow A{O}^2=B{O}^2 \\ \Rightarrow {\left(2+1\right)}^2+{\left(-3y-y\right)}^2={\left(2-5\right)}^2+{\left(-3y-7\right)}^2 \\ \Rightarrow 9+{\left(4y\right)}^2={\left(-3\right)}^2+{\left(3y+7\right)}^2 \\ \Rightarrow 9+16y^2=9+9y^2+49+42y \end{gathered}$$
$$\begin{gathered} \Rightarrow 7y^2-42y-49=0 \\ \Rightarrow y^2-6y-7=0 \\ \Rightarrow y^2-7y+y-7=0 \\ \Rightarrow y\left(y-7\right)+1\left(y-7\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow \left(y-7\right)\left(y+1\right)=0 \\ \Rightarrow y=-1\mathrm{or}y=7 \end{gathered}$$
Hence, y = 7 or y = −1.