Question

Find the ratio in which he point $(1,3)$ divides the line segment joining the points $(3,6)$ and $(-5,6)$ internally.

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

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

$P(1,3)$ divides $(3,6)$ & $(-5,6)$ in ratio , say $m:n$

Therefore, we have

$(1,3)=(\frac{-5m+3n}{m+n},\frac{6m+6n}{m+n})$

$\frac{-5m+3n}{m+n}=1$

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

$\Rightarrow 6m=2n$

$\Rightarrow \frac{m}{n}=\frac{2}{6}$

$\Rightarrow \frac{m}{n}=\frac{1}{3}$

$\Rightarrow m:n=1:3$