Question

Find the coordinates of the point which divides, internally and externally, the line joining $(-1,2)$ to $(4,-5)$ in the ratio $2:3$.

Answer 1

The coordinates of point when ${(x}_1,y_1)$ and ${(x}_2,y_2)$ are divided in $m:n$

$\left(i\right)$ internally is $(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

$\left(ii\right)$ externally $(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n})$

Let the point be $P(h,k)$

(1) For internal division

$h=\frac{2\left(4\right)+3(-1)}{2+3}=\frac{5}{5}=1k=\frac{2(-5)+3\left(2\right)}{2+3}=\frac{-4}{5}\Rightarrow P(1,\frac{-4}{5})$

(2) For external division

$h=\frac{2\left(4\right)-3(-1)}{2-3}=\frac{8+3}{-1}=-11k=\frac{2(-5)-3\left(2\right)}{2-3}=\frac{-10-6}{-1}=16\Rightarrow P(-11,16)$