Question

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

Answer 1

Suppose the line $x+y=4$ divides the join of $A(-1,1)$ and $B(5,7)$ in the ratio $\lambda :1$.

The coordinates of the point of division are $(\frac{5\lambda +1}{\lambda +1},\frac{7\lambda +1}{\lambda +1})$. It lies on $x+y=4$.

Therefore,

$\frac{5\lambda -1}{\lambda +1}+\frac{7\lambda +1}{\lambda +1}=4$

$5\lambda -1+7\lambda +1=4(\lambda +1)$

$12\lambda =4\lambda +4$

$8\lambda =4$

$\lambda =\frac{1}{2}$

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