Question

Find the ratio in which the line segment joining the points $A(3,-6)$ and $B(5,3)$ is divided by x axis.

Answer 1

Equation of line joining points $A(3,-6)$ and $B(5,3)$

$y-y_1=\frac{y_2-y_1}{x_2-x_1}\times (x-x_1)$

$⟹y+6=\frac{3+6}{5-2}(x-3)$

$⟹3x-y=15$

Point of intersection with $x-$axis is

$3x-0=15⟹x=5$

$(5,0)$

Let the ratio be $\lambda :1$

$\frac{m{x}_2+n{x}_1}{m+n}=5⟹\frac{3\lambda +1\lambda }{\lambda +1}=5$

$⟹3\lambda +5=5\lambda +5$

Also $\frac{m{y}_2+n{y}_1}{m+n}=0$

$⟹\frac{3\lambda -6\times 1}{\lambda +1}=0⟹\lambda =2$

Ratio is $\lambda :1=2:1$