Question:
If the lines \(\bar{r}=(\hat{i}-\hat{j}+2\hat{k})+\lambda (\hat{i}+2\hat{k})\) and \(\frac{x-p}{2}=y+2=z+1\) intersect, then the value of \(p\) is ...
If the lines \(\bar{r}=(\hat{i}-\hat{j}+2\hat{k})+\lambda (\hat{i}+2\hat{k})\) and \(\frac{x-p}{2}=y+2=z+1\) intersect, then the value of \(p\) is ...
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Equate the coordinates of a general point on each line and solve for the parameters.
Updated On: Nov 12, 2026
- \(-1\)
- \(1\)
- \(-2\)
- \(2\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
Write both lines in parametric form and equate coordinates at the point of intersection.
Step 2: Line 1 and Line 2
Line 1: \((1+\lambda ,\ -1,\ 2+2\lambda )\).
Line 2: \((p+2\mu ,\ \mu -2,\ \mu -1)\).
Step 3: Solve
y: \(-1=\mu -2\), so \(\mu =1\).
z: \(2+2\lambda =0\), so \(\lambda =-1\).
Step 4: Find p
x: \(1+\lambda =0=p+2\mu =p+2\), so \(p=-2\).
Final Answer:
The value of \(p\) is \(-2\). \[ \boxed{\text{(C) }-2} \]
Write both lines in parametric form and equate coordinates at the point of intersection.
Step 2: Line 1 and Line 2
Line 1: \((1+\lambda ,\ -1,\ 2+2\lambda )\).
Line 2: \((p+2\mu ,\ \mu -2,\ \mu -1)\).
Step 3: Solve
y: \(-1=\mu -2\), so \(\mu =1\).
z: \(2+2\lambda =0\), so \(\lambda =-1\).
Step 4: Find p
x: \(1+\lambda =0=p+2\mu =p+2\), so \(p=-2\).
Final Answer:
The value of \(p\) is \(-2\). \[ \boxed{\text{(C) }-2} \]
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