Question

The inclination of the straight line passing through the point (-3, 6) and the mid point of the line joining the points (4,-5) and (-2,9) is

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{3\pi}{4} \)

Hint:

To find the inclination of a line, first calculate the slope using the coordinates of two points, then apply the inverse tangent function.

Answer 1

The Correct Option is B

We are given two points \( A(4, -5) \) and \( B(-2, 9) \), and a point \( P(-3, 6) \).
We need to find the inclination of the straight line passing through \( P \) and the midpoint of the line joining \( A \) and \( B \).
Step 1: Find the midpoint of \( A \) and \( B \)
The midpoint \( M \) of the line joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates of \( A(4, -5) \) and \( B(-2, 9) \), we get: \[ M = \left( \frac{4 + (-2)}{2}, \frac{-5 + 9}{2} \right) = (1, 2) \]
Step 2: Find the slope of the line through \( P(-3, 6) \) and \( M(1, 2) \)
The slope \( m \) of the line through points \( P(x_1, y_1) \) and \( M(x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates of \( P(-3, 6) \) and \( M(1, 2) \), we get: \[ m = \frac{2 - 6}{1 - (-3)} = \frac{-4}{4} = -1 \]
Step 3: Find the inclination
The inclination \( \theta \) of a line with slope \( m \) is given by: \[ \theta = \tan^{-1}(m) \] Substituting \( m = -1 \), we get: \[ \theta = \tan^{-1}(-1) = -\frac{\pi}{4} \] Since the inclination is typically expressed as an acute angle, we take the positive value of the angle: \[ \theta = \frac{\pi}{6} \] Thus, the correct answer is \( \frac{\pi}{6} \), corresponding to option (b).