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: Understanding the Concept
\(q\hat i-2\hat j+\hat k\) is collinear with \(4\hat i-4\hat j+2\hat k\), so the ratio of components must be equal.

Step 2: Key Formula or Approach
\(\dfrac q4=\dfrac{-2}{-4}=\dfrac12\), so \(q=2\).

Step 3: Detailed Explanation
Line 1: \((1+2\lambda,\ 1-2\lambda,\ -1+\lambda)\). Line 2: \((p+\mu,\ -3-2\mu,\ 2+2\mu)\).
y: \(1-2\lambda=-3-2\mu\Rightarrow\lambda=\mu+2\).
z: \(-1+\lambda=2+2\mu\Rightarrow-1+\mu+2=2+2\mu\Rightarrow\mu=-1\), \(\lambda=1\).
x: \(1+2=p+(-1)\Rightarrow p=4\).
So \(p=4\), \(q=2\).

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