Question

If the lines \(\overset{⃗}{r} = \hat{i}+\hat{j}-\hat{k}+λ(q\hat{i}-2\hat{j}+\hat{k})\) and \(\overset{⃗}{r} = p\hat{i}-3\hat{j}+2\hat{k}+μ(\hat{i}-2\hat{j}+2\hat{k})\) intersect each other and \(q\hat{i}-2\hat{j}+\hat{k}\) is collinear to \(4\hat{i}-4\hat{j}+2\hat{k}\), then the values of \(p\) and \(q\) are

• A. \(p = 4, q = 3\)
• B. \(p = 2, q = 3\)
• C. \(p = 4, q = 2\)
• D. \(p = 4, q = 1\)

Hint:

Collinearity fixes \(q\), and intersection gives \(p\).

Answer 1

The Correct Option is C

Step 1: Use q first
Direction $(q,-2,1)$ must be a multiple of $(4,-4,2)$, i.e. half of it, giving $(2,-2,1)$, so $q=2$.

Step 2: Intersection
Equating y and z coordinates gives $\lambda=1,\mu=-1$; the x equation $3=p-1$ yields $p=4$. Option (C).

Final Answer:
$p=4,q=2$, option (C). \[ \boxed{p=4,\ q=2} \]