Question

Find the coordinates of the point which divides, internally and externally, the line joining $(-3,-4)$ to $(-8,7)$ in the ratio $7:5$.

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

$h=\frac{7(-8)+5(-3)}{7+5}=\frac{-56-15}{12}=-\frac{71}{12}=-5\frac{11}{12}k=\frac{7\left(7\right)+5(-4)}{7+5}=\frac{49-20}{12}=\frac{29}{12}=2\frac{5}{12}\Rightarrow P(-5\frac{11}{12},2\frac{5}{12})$

(2) For external division

$h=\frac{7(-8)-5(-3)}{7-5}=\frac{-56+15}{2}=-\frac{41}{2}=-20\frac{1}{2}k=\frac{7\left(7\right)-5(-4)}{7-5}=\frac{49+20}{2}=\frac{69}{2}=34\frac{1}{2}\Rightarrow P(-20\frac{1}{2},34\frac{1}{2})$