(a) The line segment joining the points A (4,-5) and B (4,5) is divided by the point P such that AP: AB =2: 5. Find the coordinates of P.

\[\begin{aligned} & \mathrm{AP}: \mathrm{AB}=2: 5 \Rightarrow \mathrm{AP}: \mathrm{PB}=2: 3 \\ & \mathrm{~A}(4,-5) \\ & \mathrm{x}=\frac{8+12}{5}=4, \mathrm{y}=\frac{10-15}{5}=-1 \\ & \text { Point } \mathrm{P} \text { is }(4,-1) \end{aligned} \]

\[\begin{aligned} & \mathrm{PA}^{2}=\mathrm{PB}^{2} \Rightarrow(\mathrm{x}-5)^{2}+(\mathrm{y}-1)^{2}=(\mathrm{x}-1)^{2}+(\mathrm{y}-5)^{2} \\ & \Rightarrow \mathrm{x}=\mathrm{y} \end{aligned} \]