Question

In what ratio the line segment joining $(-3,-1)$ and $(-8,-9)$ is divided by the point $(-5,\frac{-21}{5})$?

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

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

Here, $x_1=-3,y_1=-1,x_2=-8,y_2=-9$

$=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})=(-5,\frac{-21}{5})$

$\Rightarrow$$\frac{m(-8)+n(-3)}{m+n}=-5$

$\Rightarrow -8m-3n=-5m-5n$

$\Rightarrow -8m+5m=3n-5n$

$\Rightarrow -3m=-2n$

$\Rightarrow \frac{m}{n}=\frac{2}{3}$

$\Rightarrow m:n=2:3$

Hence, the point $(-5,\frac{-21}{5})$ divides the line segment in the ratio $2:3$.