(a) Find the ratio in which the point P (-4,6) divides the line segment joining the points A(-6,10) and B(3, -8).

Let the ratio be k:$\displaystyle 1$

\[\begin{aligned} & -4=\frac{3 k-6}{k+1} \\ & \Rightarrow k=\frac{2}{7} \end{aligned} \]

Let A$\displaystyle (3,0)$, B$\displaystyle (6,4)$, C$\displaystyle (-1, 3)$

\[\begin{aligned} & \mathrm{AB}=\sqrt{(3-6)^{2}+(0-4)^{2}}=5 \\ & \mathrm{BC}=\sqrt{(6+1)^{2}+(4-3)^{2}}=\sqrt{50} \\ & \mathrm{CA}=\sqrt{(3+1)^{2}+(0-3)^{2}}=5 \\ & \quad \mathrm{As}, \mathrm{AB}=\mathrm{AC} \\ & \Rightarrow \mathrm{ABC} \text { is an isosceles triangle } \end{aligned} \]