Question

Equation of the line parallel to the line \( \frac{x-2}{2} = \frac{y-2}{3} = \frac{z-1}{-2} \) and passing through the point \( (3, 2, -1) \) is:

• A. \( \frac{x-3}{2} = \frac{y-2}{3} = \frac{z+1}{2} \)
• B. \( \frac{x+3}{2} = \frac{y+2}{3} = \frac{z-1}{-2} \)
• C. \( \frac{x-3}{2} = \frac{y-2}{3} = \frac{z-1}{-2} \)
• D. \( \frac{x-3}{2} = \frac{y-2}{3} = \frac{z+1}{2} \)
• E. \( \frac{x+3}{2} = \frac{y+2}{3} = \frac{z+1}{-2} \)

Hint:

To find the equation of a line passing through a point and parallel to a line, use the point-direction form of the line equation.

Answer 1

The Correct Option is D

Step 1: The direction ratios of the given line are \( (2, 3, -2) \). Since the line is parallel to the given line, the direction ratios of the required line will be the same, i.e., \( (2, 3, -2) \). 
Step 2: The required line passes through the point \( (3, 2, -1) \), so we use the general form of the equation of a line passing through a point and parallel to a given direction ratios: \[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}, \] where \( (x_1, y_1, z_1) \) is the point on the line, and \( (a, b, c) \) are the direction ratios. 
Substituting \( (x_1, y_1, z_1) = (3, 2, -1) \) and \( (a, b, c) = (2, 3, -2) \): \[ \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z + 1}{2}. \]