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 Question:
We need to find the inclination angle ($\theta$) of a specific straight line.
The line passes through a given point $P(-3, 6)$ and the midpoint $M$ of a line segment connecting two other given points.

Step 2: Key Formula or Approach:
1. The midpoint $M(x,y)$ of a segment connecting $(x_1, y_1)$ and $(x_2, y_2)$ is given by $M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)$.
2. The slope $m$ of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
3. The inclination angle $\theta$ is related to the slope by $m = \tan \theta$, where $0 \le \theta < \pi$.

Step 3: Detailed Explanation:
First, find the coordinates of the midpoint $M$ of $(4, -5)$ and $(-2, 9)$:
$$M = \left( \frac{4 + (-2)}{2}, \frac{-5 + 9}{2} \right) = \left( \frac{2}{2}, \frac{4}{2} \right) = (1, 2)$$
Now, find the slope $m$ of the line passing through the given point $P(-3, 6)$ and the midpoint $M(1, 2)$:
$$m = \frac{2 - 6}{1 - (-3)} = \frac{-4}{1 + 3} = \frac{-4}{4} = -1$$
Set the slope equal to $\tan \theta$:
$$\tan \theta = -1$$
Since the inclination angle $\theta$ must fall in the interval $[0, \pi)$ and the tangent is negative, the angle lies in the second quadrant.
$$\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$

Step 4: Final Answer:
The inclination is $3\pi/4$, which matches option (d).