Question

The inclination of the straight line passing through the point $\left(-3,6\right)$ and the midpoint of the line joining the $\left(4,-5\right)$ and $\left(-2,9\right)$ is

• A. $\frac{\mathrm{\pi }}{4}$
• B. $\frac{\mathrm{\pi }}{6}$
• C. $\frac{\mathrm{\pi }}{3}$
• D. $\frac{3\mathrm{\pi }}{4}$

Answer 1

The correct option is D

$\frac{3\mathrm{\pi }}{4}$

Explanation for the correct answer:

Let $A=\left(4,-5\right)=\left(x_1,y_1\right),B\left(-2,9\right)=\left(x_2,y_2\right)$

The midpoint of $AB$ is given as

$C=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)=\left(\frac{4-2}{2},\frac{-5+9}{2}\right)$

$\Rightarrow$ $C=\left(1,2\right)$

Let $D=\left(-3,6\right)$

The slope of the line $CD$ is given as

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-2}{-3-1}$

$\Rightarrow$ $m=-1$

The slope of the line is also given as $\tan \theta$ where $\theta$ is angle made by line with positive $x$ axis

$\tan \theta =-1$

$\Rightarrow$ $\theta ={\tan }^{-1}\left(-1\right)={135}^{\circ }$

${135}^{\circ }=135\times \frac{\mathrm{\pi }}{180}$

$=\frac{3\mathrm{\pi }}{4}$ radians.

Hence, option $\left(D\right)$, $\frac{3\mathrm{\pi }}{4}$ is the correct answer.