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

$\begin {array} {1 1} (A)\;2:1 & \quad (B)\;1:2 \\ (C)\;1:3 & \quad (D)\;3:1 \end {array}$

Toolbox:
• Equation of a line joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\large\frac{y-y_1}{y_2-y_1}$$= \large\frac{x-x_1}{x_2-x_1} • Section formula : \large\frac{mx_2+nx_1}{m+n}$$, \large\frac{my_2+ny_1}{m+n}$
The given points are (-1, 1) and (5,7).
Hence equation of the line joining the above points is
$\large\frac{y-1}{7-1}$$= \large\frac{x-(-1)}{5-(-1)} \Rightarrow \large\frac{y-1}{6}$$ = \large\frac{x+1}{6}$
$\Rightarrow y-1 = x+1$
or $x-y+2=0$---------(1)
Equation of the given line is
$x+y-4=0$---------(2)
The point of intersection of the lines (1) and (2) is
$\qquad x-y=-2$
$\qquad x+y=4$
$\qquad 2x \quad = 2$
$\Rightarrow x = 1$ and $y = 3$
Let the point (1,3) divide the line segment joining (-1,1) and (5,7) in the ratio 1 : k
By applying the section formula,
$1 = \large\frac{k(-1)+1(5)}{k+1}$
$\Rightarrow k +1 = -k+5$
$\Rightarrow 2k=4$
or $k = 2$
Hence the line joining the points (-1, 1) and (5,7) is divided by line
$x+y=4$ in the ratio 1 : 2