Question:
Assertion (A) : Lines given by \( x = py + q, z = ry + s \) and \( x = p'y + q', z = r'y + s' \) are perpendicular to each other when \( pp' + rr' + 1 = 0 \).
Reason (R) : Two lines \( \vec{r} = \vec{a}_1 + \lambda\vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu\vec{b}_2 \) are perpendicular to each other if \( \vec{b}_1 \cdot \vec{b}_2 = 0 \).
Assertion (A) : Lines given by \( x = py + q, z = ry + s \) and \( x = p'y + q', z = r'y + s' \) are perpendicular to each other when \( pp' + rr' + 1 = 0 \).
Reason (R) : Two lines \( \vec{r} = \vec{a}_1 + \lambda\vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu\vec{b}_2 \) are perpendicular to each other if \( \vec{b}_1 \cdot \vec{b}_2 = 0 \).
Reason (R) : Two lines \( \vec{r} = \vec{a}_1 + \lambda\vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu\vec{b}_2 \) are perpendicular to each other if \( \vec{b}_1 \cdot \vec{b}_2 = 0 \).
Show Hint
When a line is given as \( x = ay+b, z = cy+d \), the direction ratios are \( (a, 1, c) \).
If denominators in symmetric form are \( l, m, n \), the direction vector is \( l\hat{i} + m\hat{j} + n\hat{k} \).
If denominators in symmetric form are \( l, m, n \), the direction vector is \( l\hat{i} + m\hat{j} + n\hat{k} \).
Updated On: Sep 10, 2026
- Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true and Reason (R) is false.
- Assertion (A) is false and Reason (R) is true.
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The Correct Option is A
Solution and Explanation
Concept:
• The direction of a line is determined by the vector \( \vec{b} \) in the form \( \vec{r} = \vec{a} + \lambda\vec{b} \).
• Two lines are perpendicular if the dot product of their direction vectors is zero.
Step 1: Find the direction vectors of the lines in Assertion (A)
For the first line: \( x = py + q \implies \frac{x - q}{p} = y \) and \( z = ry + s \implies \frac{z - s}{r} = y \). The symmetric form is: \[ \frac{x - q}{p} = \frac{y - 0}{1} = \frac{z - s}{r} \] The direction vector is \( \vec{b}_1 = p\hat{i} + 1\hat{j} + r\hat{k} \). For the second line: \( x = p'y + q' \implies \frac{x - q'}{p'} = y \) and \( z = r'y + s' \implies \frac{z - s'}{r'} = y \). The symmetric form is: \[ \frac{x - q'}{p'} = \frac{y - 0}{1} = \frac{z - s'}{r'} \] The direction vector is \( \vec{b}_2 = p'\hat{i} + 1\hat{j} + r'\hat{k} \).
Step 2: Apply the perpendicularity condition
The lines are perpendicular if \( \vec{b}_1 \cdot \vec{b}_2 = 0 \). \[ (p\hat{i} + \hat{j} + r\hat{k}) \cdot (p'\hat{i} + \hat{j} + r'\hat{k}) = 0 \] \[ (p)(p') + (1)(1) + (r)(r') = 0 \] \[ pp' + rr' + 1 = 0 \] Thus, Assertion (A) is True.
Step 3: Evaluate the Reason (R) and the link
Reason (R) states the general vector condition for perpendicularity, \( \vec{b}_1 \cdot \vec{b}_2 = 0 \). This is True. Since we derived the result in Step 2 directly from this principle, the Reason correctly explains the Assertion.
• The direction of a line is determined by the vector \( \vec{b} \) in the form \( \vec{r} = \vec{a} + \lambda\vec{b} \).
• Two lines are perpendicular if the dot product of their direction vectors is zero.
Step 1: Find the direction vectors of the lines in Assertion (A)
For the first line: \( x = py + q \implies \frac{x - q}{p} = y \) and \( z = ry + s \implies \frac{z - s}{r} = y \). The symmetric form is: \[ \frac{x - q}{p} = \frac{y - 0}{1} = \frac{z - s}{r} \] The direction vector is \( \vec{b}_1 = p\hat{i} + 1\hat{j} + r\hat{k} \). For the second line: \( x = p'y + q' \implies \frac{x - q'}{p'} = y \) and \( z = r'y + s' \implies \frac{z - s'}{r'} = y \). The symmetric form is: \[ \frac{x - q'}{p'} = \frac{y - 0}{1} = \frac{z - s'}{r'} \] The direction vector is \( \vec{b}_2 = p'\hat{i} + 1\hat{j} + r'\hat{k} \).
Step 2: Apply the perpendicularity condition
The lines are perpendicular if \( \vec{b}_1 \cdot \vec{b}_2 = 0 \). \[ (p\hat{i} + \hat{j} + r\hat{k}) \cdot (p'\hat{i} + \hat{j} + r'\hat{k}) = 0 \] \[ (p)(p') + (1)(1) + (r)(r') = 0 \] \[ pp' + rr' + 1 = 0 \] Thus, Assertion (A) is True.
Step 3: Evaluate the Reason (R) and the link
Reason (R) states the general vector condition for perpendicularity, \( \vec{b}_1 \cdot \vec{b}_2 = 0 \). This is True. Since we derived the result in Step 2 directly from this principle, the Reason correctly explains the Assertion.
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