Question

If the straight lines \( \frac{x - 3}{2} = \frac{y - 4}{3} = \frac{z - 6}{-1} \) and \( \frac{x - 2}{a} = \frac{y + 3}{b} = \frac{z + 4}{-1} \) are parallel, then \( a^2 + b^2 \) is:

• A. 1
• B. 13
• C. 24
• D. 17
• E. 3

Hint:

For parallel lines, the direction ratios are proportional. Use this property to solve for \( a \) and \( b \).

Answer 1

The Correct Option is B

The lines are parallel if their direction ratios are proportional. The direction ratios of the first line are \( \langle 2, 3, -1 \rangle \), and the direction ratios of the second line are \( \langle a, b, -1 \rangle \). For the lines to be parallel, we must have: \[ \frac{2}{a} = \frac{3}{b} = \frac{-1}{-1} \] Thus, \( \frac{2}{a} = \frac{3}{b} = 1 \), which gives: \[ a = 2 \quad {and} \quad b = 3 \] Now, calculate \( a^2 + b^2 \): \[ a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 \]