Question

Find the ratio in which $(4,5)$ divides the line segment joining of $(2,3)$ and $(7,8)$.

Answer 1

Using the section formula, if a point $(x,y)$ divides the line segment 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, the point $(4,5)$ divides the line segment joining the points $(2,3)$ and $(7,8)$

Let the ratio be $\lambda :1$
we have
$(\frac{7\times \lambda +2\times 1}{\lambda +1},\frac{8\times \lambda +3\times 1}{\lambda +1})=(4,5)$

$\therefore \frac{7\lambda +2}{\lambda +1}=4$

$\Rightarrow 7\lambda +2=4\lambda +4$

$\Rightarrow 7\lambda -4\lambda =4-2$

$\Rightarrow 3\lambda =2$

$\Rightarrow \lambda =\frac{2}{3}$

$\therefore$ Required ratio $=\frac{2}{3}:1=2:3$