Question

The midpoint of the line joining the points $(-10,8)$ and $(-6,12)$ divides the line joining the points $(4,-2)$ and $(-2,4)$ in the ratio

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

Answer 1

The correct option is D

$2:1$ externally

Explanation for the correct option:

Step 1: Find the midpoint of $(-10,8)$ and $(-6,12)$.

According to the midpoint formula, the midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ is given by $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Therefore, the midpoint of $(-10,8)$ and $(-6,12)$ is $\left(\frac{-10-6}{2},\frac{8+12}{2}\right)=\left(-8,10\right)$.

Step 2: Find the required ratio.

Since, it is given that $(-8,10)$ divides the line joining the point $(4,-2)$ and $(-2,4)$.

Find the ratio in which $(-8,10)$ divides the line joining the point $(4,-2)$ and $(-2,4)$.

Assume that the required ration is $m:n$.

Therefore, according to the section formula:

$$\begin{gathered} \frac{m(-2)+n\left(4\right)}{m+n}=-8 \\ \Rightarrow -2m+4n=-8m-8n \\ \Rightarrow 6m=-12n \\ \Rightarrow \frac{m}{n}=\frac{-12}{6} \\ \Rightarrow \frac{m}{n}=\frac{-2}{1} \end{gathered}$$

Therefore, $(-8,10)$ divides the line joining the point $(4,-2)$ and $(-2,4)$ in $2:1$and the negative sign represents the external division.

Hence, option $D$ is the correct answer.