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

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- 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

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