Question

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

Answer 1

The given line is $x+y=3$ ..........$\left(1\right)$

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 $\frac{k+3}{k+1}+\frac{3k+2}{k+1}=3$

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

$\Rightarrow 4k+5=3k+3$

$\Rightarrow k=-2$

$\Rightarrow k=\frac{-2}{1}$

Hence, the required ratio is $2:1$ externally.