# Q : 14     In what ratio, the line joining  $\small (-1,1)$  and  $\small (5,7)$  is divided by the line  $x+y=4$ ?

Equation of line joining  $\small (-1,1)$  and  $\small (5,7)$ is
$(y-1)= \frac{7-1}{5+1}(x+1)$
$\Rightarrow (y-1)= \frac{6}{6}(x+1)$
$\Rightarrow (y-1)= 1(x+1)$
$\Rightarrow x-y+2=0$
Now, point of intersection of lines $x+y=4$  and $x-y+2=0$    is   $(1,3)$
Now, let's suppose point $(1,3)$divides the line  segment   joining  $\small (-1,1)$  and  $\small (5,7)$   in  $1:k$
Then,
$(1,3)= \left ( \frac{k(-1)+1(5)}{k+1},\frac{k(1)+1(7)}{k+1} \right )$
$1=\frac{-k+5}{k+1} \ \ and \ \ 3 = \frac{k+7}{k+1}$
$\Rightarrow k =2$
Therefore, the line joining  $\small (-1,1)$  and  $\small (5,7)$  is divided by the line  $x+y=4$  in ratio $1:2$

Exams
Articles
Questions