Question:
Direction ratios of lines \( l_1 \) and \( l_2 \) respectively are \( \langle 1, -2, 3 \rangle \) and \( \langle -2, p, -6 \rangle \). The value of p for which \( l_1 \parallel l_2 \), is :
Direction ratios of lines \( l_1 \) and \( l_2 \) respectively are \( \langle 1, -2, 3 \rangle \) and \( \langle -2, p, -6 \rangle \). The value of p for which \( l_1 \parallel l_2 \), is :
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Parallel lines possess direction vectors that are scalar multiples of each other. Inspecting the numbers here, multiplying line 1's vector by \(-2\) directly yields line 2's vector: \(-2 \times (-2) = 4\).
Updated On: Jul 4, 2026
- \( -4 \)
- \( 4 \)
- \( -10 \)
- \( 10 \)
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The Correct Option is B
Solution and Explanation
Concept:
For two lines in three-dimensional space to be parallel, their direction ratios must be proportional to one another. If the direction ratios of the first line are \( \langle a_1, b_1, c_1 \rangle \) and the second line are \( \langle a_2, b_2, c_2 \rangle \), then the condition for parallelism is:
\[
\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
\]
Step 1: Extract the components and set up the proportion.
Given direction ratios: - Line 1: \( a_1 = 1, \, b_1 = -2, \, c_1 = 3 \) - Line 2: \( a_2 = -2, \, b_2 = p, \, c_2 = -6 \) Applying the condition for parallel lines: \[ \frac{1}{-2} = \frac{-2}{p} = \frac{3}{-6} \]
Step 2: Simplify the known ratios and solve for \(p\).
Notice that the first and third ratios both simplify to \( -\frac{1}{2} \): \[ -\frac{1}{2} = \frac{-2}{p} \] Cross-multiplying to solve for \( p \): \[ 1 \cdot p = (-2) \cdot (-2) \] \[ p = 4 \] This value corresponds to option (B).
Step 1: Extract the components and set up the proportion.
Given direction ratios: - Line 1: \( a_1 = 1, \, b_1 = -2, \, c_1 = 3 \) - Line 2: \( a_2 = -2, \, b_2 = p, \, c_2 = -6 \) Applying the condition for parallel lines: \[ \frac{1}{-2} = \frac{-2}{p} = \frac{3}{-6} \]
Step 2: Simplify the known ratios and solve for \(p\).
Notice that the first and third ratios both simplify to \( -\frac{1}{2} \): \[ -\frac{1}{2} = \frac{-2}{p} \] Cross-multiplying to solve for \( p \): \[ 1 \cdot p = (-2) \cdot (-2) \] \[ p = 4 \] This value corresponds to option (B).
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