Question

Determine the ratio in which the point $P(m,6)$ divides the line segment, of $A(-4,3)$ and $B(2,8)$. Also, find the value of $m$.

Answer 1

Let point P divides $AB$ in ratio $\lambda :1$

Section Formula: If point $(x,y)$ divides line joining two points in

$(x_1,y_1)$ and $(x_2,y_2)$ in ratio $m:n$ internally

$x=\frac{m{x}_2+n{x}_1}{m+n}$

$m=\frac{2\lambda -4}{\lambda +1}$$=\frac{2\times \frac{3}{2}-4}{\frac{3}{2}+1}$ $=\frac{-2}{5}$

Now, $y=\frac{m{y}_2+n{y}_1}{m+n}$

$6=\frac{8\lambda +3}{\lambda +1}$

$6\lambda +6=8\lambda +3$

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

Ratio, $\lambda :1$ $=\frac{3}{2}:1$ $=3:2$

$m=\frac{-2}{5}$