A point P divides the line segment joining the points $\displaystyle \mathrm{A}(-3,5)$ and B$\displaystyle (7, -4)$ in a certain ratio. If the point P lies on the line $\displaystyle \mathrm{y}=2 \mathrm{x}$, then find the ratio AP : PB and coordinates of point P.

Let AP : PB = k : $\displaystyle 1$ Coordinates of point P are \(\displaystyle \left(\frac{7 \mathrm{k}-3}{\mathrm{k}+1}, \frac{-4 \mathrm{k}+5}{\mathrm{k}+1}\right)\) Since point P lies on the line \(\displaystyle \mathrm{y}=2 \mathrm{x}\) Therefore, \(\displaystyle \frac{-4 \mathrm{k}+5}{\mathrm{k}+1}=2 \times\left(\frac{7 \mathrm{k}-3}{\mathrm{k}+1}\right)\) \[\Rightarrow \mathrm{k}=\frac{11}{18} \] \(\displaystyle \therefore \mathrm{AP}: \mathrm{PB}=11: 18\) Coordinates of P are \(\displaystyle \left(\frac{7 \times 11-3 \times 18}{11+18}, \frac{-4 \times 11+5 \times 18}{11+18}\right)=\left(\frac{23}{29}, \frac{46}{29}\right)\)