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: First line
\(2x = ky = -z = t\) gives \(x = \frac t2\), \(y = \frac tk\), \(z = -t\). Direction ratios \(\left(\frac12,\frac1k,-1\right)\), or multiplying by \(2k\), \((k,2,-2k)\).

Step 2: Second line
\(6x = -y = -4z = s\) gives \(x = \frac s6\), \(y = -s\), \(z = -\frac s4\). Multiply by \(12\): \((2,-12,-3)\).

Step 3: Perpendicular condition
\(k\cdot2 + 2(-12) + (-2k)(-3) = 0\) gives \(2k - 24 + 6k = 0\), so \(8k = 24\) and \(k=3\). Option (D).

Final Answer:
The value of k is 3. \[ \boxed{\text{(D)}\ 3} \]