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 the coordinates of the point of division.

Answer 1

Solution-

Line joining $(3,-1)$ & $(8,9)$ is

$y+1=\frac{9+1}{8-3}(x-3)$

$y+1=2x-6$

$y=2x-7$

Find intersection with $x-y=2\Rightarrow y=x-2$

$x-2=2x-7\Rightarrow x=5$

$y=5-2=3$

the intersecting point $P(5,3)$

Let this point divides line in ratio $m:n$

then $(\frac{8m+3n}{m+n},\frac{9m-n}{m+n})=(5,3)$ [section formula]

$\frac{8m+3n}{m+n}=5$

$8m+3n=5m+5n$

$3m=2n$

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

$m:n=3:2$