(a) Find a relation between x and y such that the point P (x, y ) is equidistant from the points A(7, 1) and B(3, 5).

\[\begin{aligned} & \mathrm{PA}=\mathrm{PB} \\ & \Rightarrow \mathrm{PA}^{2}=\mathrm{PB}^{2} \\ &(\mathrm{x}-7)^{2}+(\mathrm{y}-1)^{2}=(\mathrm{x}-3)^{2}+(\mathrm{y}-5)^{2} \\ & \Rightarrow-8 \mathrm{x}+8 \mathrm{y}+16=0 \text { or } \mathrm{x}-\mathrm{y}-2=0 \end{aligned} \]

Centre O ($\displaystyle 2$, - 3y) is the mid point of AB

\[\begin{aligned} & \therefore \frac{y+7}{2}=-3 y \\ & \Rightarrow y=-1 \end{aligned} \]

Radius \(\displaystyle =\mathrm{OB}=\sqrt{(5-2)^{2}+(7-3)^{2}}=5\)