Question

The equation of a line passing through the point \( (2, 4, 6) \) and parallel to the line \[ 3x + 4 = 4y - 1 = 1 - 4z \] is

• A. \( \frac{x - 2}{4} = \frac{y - 4}{3} = \frac{z - 6}{3} \)
• B. \( \frac{x - 2}{4} = \frac{y - 4}{3} = \frac{z - 6}{-3} \)
• C. \( \frac{x - 2}{-4} = \frac{y - 4}{3} = \frac{z - 6}{3} \)
• D. \( \frac{x - 2}{-4} = \frac{y - 4}{-3} = \frac{z - 6}{-3} \)

Hint:

For a line parallel to another, use the direction ratios of the original line and the point the line passes through to write the equation.

Answer 1

The Correct Option is B

Step 1: Find the direction ratios of the given line.
The direction ratios of the given line are the coefficients of \( x \), \( y \), and \( z \) in the equation \( 3x + 4 = 4y - 1 = 1 - 4z \). These direction ratios are: \[ (3, 4, -4) \]
Step 2: Use the point and direction ratios to write the equation of the line.
For the line passing through the point \( (2, 4, 6) \) and parallel to the given line, the equation is: \[ \frac{x - 2}{4} = \frac{y - 4}{3} = \frac{z - 6}{-3} \]
Step 3: Conclusion.
Thus, the equation of the line is \( \frac{x - 2}{4} = \frac{y - 4}{3} = \frac{z - 6}{-3} \), corresponding to option (B).