Question

Find the coordinates of a point P, which lies on the line segment joining the points A (− 2, − 2) and B (2, − 4) such that $\mathrm{AP}=\frac{3}{7}\mathrm{AB}$

Answer 1

$$\begin{gathered} \mathrm{AP}=\frac{3}{7}\mathrm{AB} \\ \Rightarrow \frac{\mathrm{AB}}{\mathrm{AP}}=\frac{7}{3} \\ \Rightarrow \frac{\mathrm{AB}}{\mathrm{AP}}-1=\frac{7}{3}-1 \\ \Rightarrow \frac{\mathrm{AB}-\mathrm{AP}}{\mathrm{AP}}=\frac{7-3}{3} \\ \Rightarrow \frac{\mathrm{PB}}{\mathrm{AP}}=\frac{4}{3} \\ \Rightarrow \frac{\mathrm{AP}}{\mathrm{PB}}=\frac{3}{4} \end{gathered}$$
So, the point P divides the line segment AB in the ratio 3 : 4.
Let the coordinates of P be (x, y)
$\therefore x=\frac{3\times 2+4\left(-2\right)}{3+4}=\frac{-2}{7}\mathrm{and}y=\frac{3\left(-4\right)+4\left(-2\right)}{3+4}=\frac{-20}{7}$
Hence, the coordinates of the point P is $\left(\frac{-2}{7},\frac{-20}{7}\right)$.