Question

The ratio in which the line $x+y=4$divides the line joining the points $(-1,1)$ and $\left(5,7\right)$ is

• A. $1:2$
• B. $2:1$
• C. $1:3$
• D. $3:1$
• E. $3:2$

Answer 1

The correct option is A

$1:2$

Explanation of the correct option:

Step $1$: Find the intersection point.

Given points are $(-1,1)$ and $\left(5,7\right)$.

Therefore, equation of the line joining the above points is

$$\begin{gathered} \frac{y-1}{7-1}=\frac{x-(-1)}{5-(-1)} \\ \Rightarrow \frac{y-1}{6}=\frac{x-(-1)}{6} \\ \Rightarrow y-1=x+1 \\ \Rightarrow x-y+2=0.......................\left(1\right) \end{gathered}$$

Equation of the given line is

$x+y-4=0$…………………$\left(2\right)$

The point of intersection of the lines $\left(1\right)$ and $\left(2\right)$ is,

$$\begin{gathered} x-y=-2 \\ x+y=4 \\ \Rightarrow 2x=2 \\ \Rightarrow x=1 \\ \Rightarrow y=3 \end{gathered}$$

Step $2$: Compute the ratio.

Let the point $\left(1,3\right)$ divide the line segment joining $(-1,1)$ and $\left(5,7\right)$ in the ratio $1:\mathrm{k}$.

Apply the section formula,

$$\begin{gathered} 1=\frac{\mathrm{k}(-1)+\left(1\right)\left(5\right)}{\mathrm{k}+1} \\ \Rightarrow \mathrm{k}+1=-\mathrm{k}+5 \\ \Rightarrow 2\mathrm{k}=4 \\ \Rightarrow \mathrm{k}=2 \end{gathered}$$

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

Hence option (A) is the correct option.