Question

In what ratio is the line segment joining $(-2,1)$ and $(-1,7)$ divided by the line $x+y=4?$

Answer 1

The given line is $x+y=4$ ..........$\left(1\right)$

Let the line $\left(1\right)$ divide the line segment joining the point $(-2,1)$ and $(-1,7)$ at the point $P$ in the ratio $k:1$ then the co-ordinates of point $P$ are

$(\frac{-k-2}{k+1},\frac{7k+1}{k+1})$

Since $P$ lies on $\left(1\right)$, we get $\frac{-k-2}{k+1}+\frac{7k+1}{k+1}=4$

$\Rightarrow -k-2+7k+1=4k+4$

$\Rightarrow 6k-1=4k+4$

$\Rightarrow 2k=5$

$\Rightarrow k=\frac{5}{2}$

Hence, the required ratio is $\frac{5}{2}:1$ or $5:2$ internally.