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

The equation of the line joining the points (-1,1) and (5,7) is,

\( y-1=\dfrac{7-1}{5+1}(x+1)\)

\( y-1=\dfrac{6}{6}(x+1)\)

x - y + 2 = 0............(1)

The equation of the given line is x + y -4 = 0..............(2) .

The points of intersection of line (1) and (2) is x =1 and y = 3.

Let point (1,3) divide the line segment joining (-1,1) and (5,7) in the ratio 1: k .

Then, by section formula,

\( ( 1,3)=(\dfrac{-k+5}{1+k},\dfrac{k+7}{1+k})\)

\( \dfrac{-k+5}{1+k}=1,\)\( \dfrac{k+7}{1+k}=3\)

\( \dfrac{-k+5}{1+k}=1\)

By cross multiplicatio

= -k+5=1+k

= 2k =4

k =2

Therefore, the line joining the points (-1,1) and (5,7) is divided by line x + y = 4 in the ratio 1: 2.