Question

If lines $\frac{x - 1}{-3} = \frac{y - 2}{2k} = \frac{z - 3}{2}$ and $\frac{x - 1}{3k} = \frac{y - 5}{1} = \frac{z - 6}{-5}$ are mutually perpendicular, then $k$ is equal to:

• A. $-\frac{10}{7}$
• B. $\frac{7}{10}$
• C. $-10$
• D. $-7$

Answer 1

The Correct Option is A

The direction ratios of the lines are $(-3, 2k, 2)$ and $(3k, 1, -5)$. For the lines to be perpendicular, the dot product of their direction ratios must be zero: \[ -3(3k) + 2k(1) + 2(-5) = 0. \] Simplify: \[ -9k + 2k - 10 = 0 \implies -7k = 10 \implies k = -\frac{10}{7}. \]