Question

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

Answer 1

Here P(x,y) divides line segment AB such that $AP=\frac{3}{7}AB$


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

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

$\Rightarrow \frac{AB}{AP}-1=\frac{7}{3}-1$

$\Rightarrow \frac{AB-AP}{AP}=\frac{4}{3}$

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

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

$\therefore$ P divides AB in the ratio 3:4 (m:n)

The coordinates of P are (x,y)

Therefore,

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

$x=\frac{3\times 2+4(-2)}{3+4},y=\frac{3(-4)+4(-2)}{3+4}$

$x=\frac{6-8}{7},y=\frac{-12-8}{7}$

Therefore,co-ordinates of P(x,y) are $(\frac{-2}{7},\frac{-20}{7})$.