Question

In what ratio does the line $x-y-2=0$ divide the line segment joining the points$(3,-1)$ and $(8,9)$? Also, find co-ordinates of point of intersection.

Answer 1

Let A(3,-1) and B(8,9).

Let the line $x-y-2=0$ divide the line segment AB at point C(x,y) in the ratio m:n.

By section formula,

$C(x,y)=[\frac{m\times 8+n\times 3}{m+n},\frac{m\times 9+n\times -1}{m+n}]$

$C(x,y)=[\frac{8m+3n}{m+n},\frac{9m-n}{m+n}]$

Since, point $C$ lie on the line $x-y-2=0$,

$\frac{8m+3n}{m+n}-\frac{9m-n}{m+n}-2=0$

$8m+3n-9m+n-2m-2n=0$

$3m=2n$

$\frac{m}{n}=\frac{2}{3}$

Hence, the required ratio is $2:3$.