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The line $x+y=4$ divides the line joining $\text{(-1,1) & (5,7)}$ in the ratio $\lambda : 1$ then the value of $\lambda$ is:

1. $2$
2. $3$
3. $\dfrac{1}{2}$
4. $1$

Ans is option (C)

By using the two point form of equation of a straight line,  (for the points  $(-1,1)$ and $(5,7)$)

$y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\times(x-x_{1})$

By substituting the points in the given formula, we get the equation of the straight line as  $y=x+2$

So, now the intersection point of $y=x+2$ and $x+y=4$ is $(1,3)$

Since the line $x+y=4$ divides the line $y=x+2$ in the ratio $\lambda:1$, we use the section formula to determine the value of $\lambda.$  (Ref: Section Formula)

$\therefore$  $\frac{\lambda(5)+1(-1)}{\lambda+1}=1$     $\Rightarrow$   $4\lambda=2$   $\Rightarrow$   $\lambda=\frac{1}{2}$

by
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Another method of solving the above problem.

Ratio is $\lambda :1$ and points given are (-1, 1) and (5, 7)

Therefore, point is $(\frac{5\lambda -1}{\lambda +1}, \frac{7\lambda +1}{\lambda +1})$

This point lies on the given line x + y = 4

$\frac{5\lambda -1}{\lambda +1} + \frac{7\lambda +1}{\lambda +1} =4$

On solving, we get value of ${\lambda} = \frac{1}{2}$
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