Question

The line segment joining the points A(4, −5) and B(4, 5) is divided by the point P such that $\frac{AP}{AB}=\frac{2}{5}$. Find the coordinates of P.

Answer 1

The given points are A(4, −5) and B(4, 5).
P divides AB, such that $\frac{AP}{AB}=\frac{2}{5}$.
$$\begin{gathered} \mathrm{Here},AB=AP+PB \\ \mathrm{Therefore},\frac{AP}{AP+PB}=\frac{2}{5} \\ \Rightarrow 5AP=2AP+2PB \\ \Rightarrow 5AP-2AP=2PB \\ \Rightarrow 3AP=2PB \\ \mathrm{Therefore},\frac{AP}{PB}=\frac{2}{3} \\ \mathrm{So},m=2\mathrm{and}n=3. \end{gathered}$$

$$\begin{gathered} x=\frac{\left(m{x}_2+n{x}_1\right)}{\left(m+n\right)},y=\frac{\left(m{y}_2+n{y}_1\right)}{\left(m+n\right)} \\ \Rightarrow x=\frac{\left(2\times 4+3\times 4\right)}{2+3},y=\frac{\left(2\times 5+3\times \left(-5\right)\right)}{2+3} \\ \Rightarrow x=\frac{8+12}{5},y=\frac{10-15}{5} \\ \Rightarrow x=\frac{20}{5},y=\frac{-5}{5} \\ x=4\mathrm{and}y=-1 \\ \mathrm{Hence},\text{the coordinates of}P\text{are}(4,-1). \end{gathered}$$