Question

In what ratio is the line segment joining $(-3,-2)$ and $(1,3)$ divided by the line $x-y=5?$

Answer 1

The given line is $x-y=5$ ..........$\left(1\right)$

Let the line $\left(1\right)$ divide the line segment joining the point $(-3,-2)$ and $(1,3)$ at the point $P$ in the ratio $k:1$ then the co-ordinates of point $P$ are

$(\frac{k-3}{k+1},\frac{3k-2}{k+1})$

Since $P$ lies on $\left(1\right)$, we get $\frac{k-3}{k+1}-\frac{3k-2}{k+1}=5$

$\Rightarrow k-3-3k+2=5k+5$

$\Rightarrow -2k-1=5k+5$

$\Rightarrow -7k=6$

$\Rightarrow k=\frac{-6}{7}$

Hence, the required ratio is $6:7$ externally.