Question

In what ratio is the line segment joining $(-2,1)$ and $(-1,-7)$ divided by the line $x+y=10?$

Answer 1

The given line is $x+y=10$ ..........$\left(1\right)$

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 $\frac{-k-2}{k+1}+\frac{-7k+1}{k+1}=10$

$\Rightarrow -k-2-7k+1=10k+10$

$\Rightarrow -8k-1=10k+10$

$\Rightarrow -18k=11$

$\Rightarrow k=\frac{-11}{18}$

Hence, the required ratio is $\frac{-11}{18}:1$ or $11:8$ externally.