Question

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

• A. $2:3$
• B. $3:2$
• C. $4:3$
• D. $3:4$

Answer 1

The correct option is A $2:3$

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

Let the required ratio be $k:1$

Substituting $(x_1,y_1)=(-3,-1)$ and $(x_2,y_2)=(-8,-9)$ in the
section formula, we get $(\frac{k(-8)+1(-3)}{k+1},\frac{k(-9)+1(-1)}{k+1})=(-5,-\frac{21}{5})$

$(\frac{-8k-3}{k+1},\frac{-9k-1}{k+1})=(-5,-\frac{21}{5})$

Comparing the $x$ - coordinate,
$=>\frac{-8k-3}{k+1}=-5$
$=>-8k-3=-5k-5$

$3k=2$

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

Hence, the required ratio is $2:3$.