Question

If $A$ and $B$ are $(-2,-2)$ and $(2,-4)$ respectively. Find the coordinates of $P$ such that $AP=\frac{3}{7}AB$ and $P$ lies on the segment $AB$.

Answer 1

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

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

Since, $AP=\frac{3}{7}(AP+PB)=\frac{4}{7}AP=\frac{3}{7}PB$

$\Rightarrow \frac{AP}{PB}=\frac{3}{4}$

$\Rightarrow$ By section formula, $P_x=\frac{3\left(2\right)+4(-2)}{7}=\frac{-2}{7}$

$P_y=\frac{3(-4)+4(-2)}{7}=\frac{-20}{7}$

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