Question

In what ratio, the line joining $(-1,1)$ and $(5,7)$ is divided by the line $x+y=4$?

Answer 1

The equation of the line joining the points $(-1,1)$ and $(5,7)$ is given by,
$y-1=\frac{7-1}{5+1}(x+1)$
$y-1=\frac{6}{6}(x+1)$
$x-y+2=0$......(1)
and the equation of the given line is
$x+y-4=0$.....(2)
Thus point of intersection of lines (1) and (2) is given by,
$x=1$ and $y=3$
Let point $(1,3)$ divide the line segment joining $(-1,1)$ and $(5,7)$ in the ration $1:k$.
Accordingly, by section formula,
$(1,3)=\{\frac{k(-1)+1\left(5\right)}{1+k},\frac{k\left(1\right)+1\left(7\right)}{1+k}\}$
$\Rightarrow (1,3)=\{\frac{-k+5}{1+k},\frac{k+7}{1+k}\}$
$\Rightarrow \frac{-k+5}{1+k}=1,\frac{k+7}{1+k}=3$
$\Rightarrow \frac{-k+5}{1+k}=1$
$\Rightarrow -k+5=1+k$
$\Rightarrow 2k=4\Rightarrow k=2$
Thus, the line joining the points $(-1,1)$ and $(5,7)$ is divided by line $x+y=4$ in the ratio $1:2$.