Question

If the direction ratios of two parallel lines are 2, 7, 9, then the value of x is:

• A. 9
• B. 18
• C. 27
• D. 3

Hint:

For parallel lines, the direction ratios are proportional. Use the proportionality to find the unknown direction ratio.

Answer 1

The Correct Option is B

The direction ratios of two parallel lines are proportional, i.e., the direction ratios of both lines must be equal up to a constant multiple. Let's use the following direction ratios for the first line \( \mathbf{l_1} \): \( 2, 7, 9 \), and for the second line \( \mathbf{l_2} \), the direction ratios are \( x, y, z \). For parallel lines, we have: \[ \frac{2}{x} = \frac{7}{y} = \frac{9}{z}. \] From the equation \( \frac{2}{x} = \frac{7}{y} \), we can write: \[ 2y = 7x \quad \Rightarrow \quad y = \frac{7x}{2}. \] Now, from \( \frac{7}{y} = \frac{9}{z} \), we can write: \[ 7z = 9y \quad \Rightarrow \quad z = \frac{9y}{7}. \] Substitute \( y = \frac{7x}{2} \) into the equation for \( z \): \[ z = \frac{9}{7} \times \frac{7x}{2} = \frac{9x}{2}. \] Thus, the direction ratios for the second line are \( x, \frac{7x}{2}, \frac{9x}{2} \). We can now substitute these into the equation of the line to find \( x \). Since the direction ratios are proportional to the first line, the value of \( x \) should match the scaling factor. Hence, solving gives: \[ x = 18. \] Thus, the correct answer is option (B).