Question

If $A$ and $B$ are two points having coordinates $(-2,-2)$ and $(2,-4)$ respectively, find the coordinates of $P$ such that $AP=\frac{3}{7}AB$.

Answer 1

Given:

$AP=\frac{3}{7}AB$

$\frac{AP}{AB}=\frac{3}{7}$

$\frac{AP}{AP+BP}=\frac{3}{3+4}$

$\frac{AP}{BP}=\frac{3}{4}$

Hence, the line joining the points $A(-2,-2)$ and $B(2,-4)$ is divided by $P$ in ratio $3:4$.

$P(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Substituting the values, $x_1=-2,y_1=-2,x_2=2,y_2=-4$ in the section formula.

$P=(\frac{3\left(2\right)+4(-2)}{3+4},\frac{3(-4)+4(-2)}{3+4})$

$P=(\frac{-2}{7},\frac{-20}{7})$