Question

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

Answer 1

We know that by section formula, the co-ordinates of the point which divides internally the line segment joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ is

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

Let the ratio of division be $k:1$,

then the co-ordinate of the point$=(\frac{-8k-3}{k+1},\frac{-9k-1}{k+1})$

ATQ,

$(-5,\frac{-21}{5})$

$\therefore \frac{-8k-3}{k+1}=-5\Rightarrow -8k-3=-5k-5\Rightarrow -8k+5k=-5+3$

$\Rightarrow -3k=-2$

$\therefore k=\frac{2}{3}$

Hence, the required ratio is $=\frac{2}{3}:1=2:3$