Question

If the lines \(2x = ky = -z\) and \(6x = -y = -4z\) are perpendicular to each other then the value of \(k\) is ...

• A. \(16\)
• B. \(5\)
• C. \(10\)
• D. \(3\)

Hint:

Write each line in symmetric form to read its direction ratios, then set their dot product to zero.

Answer 1

The Correct Option is D

Step 1: Use fractions directly
Direction vectors: $\vec d_1 = \left(\frac12,\frac1k,-1\right)$ and $\vec d_2 = \left(\frac16,-1,-\frac14\right)$.

Step 2: Dot product
$\frac1{12} - \frac1k + \frac14 = 0$ so $\frac1k = \frac13$ and $k=3$. Option (D).

Final Answer:
k = 3. \[ \boxed{\text{(D)}\ 3} \]