Question

In what ratio is the line joining the points $(2,3)$ and $(4,-5)$ divided by the line passing through the points $(6,8)$ and $(-3,-2)$ .

Answer 1

equation of line passing through (6,8) and (-3,-2)

$\left(\frac{y-8}{x-6}\right)=\left(\frac{8-(-2)}{6-(-3)}\right)or\left(\frac{y-8}{x-6}\right)=\left(\frac{10}{9}\right)$

$or9y-72=10x-60or9y=10x+12$

let the ratio in which line joining the points (2,3) and (4,-5) is divided is$\lambda :1$

then the point of intersection will be$p\left(\right(\frac{4\lambda +2}{\lambda +1}),(\frac{-5\lambda +3}{\lambda +1}\left)\right)$

as this points lies on line

$\therefore 9\left(\right(\frac{-5\lambda +3}{\lambda +1}\left)\right)=10\left(\right(\frac{4\lambda +2}{\lambda +1}\left)\right)+12-45\lambda +27=40\lambda +20+12(\lambda +1)-45\lambda +27=40\lambda +20+12\lambda +12$

$or-5=97\lambda$

$\therefore \lambda =\left(\frac{-5}{97}\right)\left(externaldivision\right)$