Question

The straight line passing through $(-3, 6)$ and midpoint of the line segment joining the points $(4, -5)$ and $(-2, 9)$ have inclination ______.

• A. $\pi/4$
• B. $\pi/6$
• C. $\pi/3$
• D. $3\pi/4$

Hint:

Remember that the "inclination" of a line is strictly the angle it makes with the positive direction of the x-axis, so it must always be a positive angle between $0$ and $180^\circ$ (or $0$ to $\pi$ radians).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The inclination $\theta$ of a line is related to its slope $m$ by $m = \tan \theta$. First, we find the midpoint, then the slope between the points.

Step 2: Formula Application:
Midpoint $M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)$. Slope $m = \frac{y_2-y_1}{x_2-x_1}$.

Step 3: Explanation:
Midpoint $M = \left( \frac{4-2}{2}, \frac{-5+9}{2} \right) = (1, 2)$. Line passes through $P(-3, 6)$ and $M(1, 2)$. $m = \frac{2-6}{1-(-3)} = \frac{-4}{4} = -1$. $\tan \theta = -1$. Since inclination is $0 \leq \theta < \pi$, $\theta = 135^\circ = 3\pi/4$.

Step 4: Final Answer:
The inclination is $3\pi/4$.