Given line is x+ y = 4 .............1

and points are: (-1,1) and (5,7)

Equation of line which joins the points

       y-1 = {(7-1)/(5+1)}*(x+1)

=> y-1 = (6/6)*(x+1)

=> y-1 = x+1

=> x-y + 1 +1 = 0

=> x-y+2 = 0 ..........2

After solving equation 1 and 2

x = 1, y = 3

So (1,3) is the point of intersection of these two lines.

Now let point (1,3) divide the line joining (-1,1) and (5,7) in the ratio 1:k

According to section formula,

      (1,3) = [{k(-1) + 1*5}/(1+k), (k*1 + 1*7)/(1+k)]

=> (1,3) = {(-k+5)/(1+k), (k+7)/(1+k)}

Now

        (-k+5)/(1+k) = 1 

  => -k+5 = 1+k 

  =>  5-1= k+k 

  => 4 = 2k 

  => k=2 

So the required ratio is 1:2