Question

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

• A. $2:1$
• B. $1:2$ internally
• C. $1:2$ externally
• D. none of these

Answer 1

The correct option is B

$1:2$ internally

Explanation for correct option:

Step-1 Finding intersection point:

Equation of line formed by the points $(x_1,y_2)$ and $(x_2,y_2)$ is $\left(y-y_1\right)\mathbf{=}\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\left(x-x_1\right)$

Equation of line formed by the points $(-1,1)$ and $(5,7)$ is $\left(y-1\right)=\frac{\left(7-1\right)}{\left(5+1\right)}\left(x+1\right)$

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

Adding $x-y=-2$ and $x+y=4$

$$\begin{gathered} 2x=2 \\ x=1 \end{gathered}$$

Putting value od x in equation $\left(1\right)$

$$\begin{gathered} 1-y=-2 \\ -y=-2-1 \\ y=3 \end{gathered}$$

Step-2 Finding the ratio:

Given two equation intersect at $(1,3)$

Let $(-1,1)$ be $(x_1,y_1)$ , $(5,7)$ be $(x_2,y_2)$ and $(1,3)$ be $(x,y)$

Let $(1,3)$ bisect $(-1,1)$ and $(5,7)$ in ratio $1:k$

$$\begin{gathered} (x,y)=\left(\frac{1\times x_2+k\times x_1}{1+k},\frac{1\times y_2+k\times y_1}{1+k}\right) \\ (1,3)=\left(\frac{1\times \left(5\right)+k\times (-1)}{k+1},\frac{1\times 7+k\times 1}{k+1}\right) \\ \Rightarrow k+1=5-k,3k+3=7+k \\ \Rightarrow 2k=4 \\ \Rightarrow k=2 \end{gathered}$$

Hence,option B is correct.