Question

Direction ratios of lines \(l_1\) and \(l_2\) are \(\) and \(\) respectively. The values of \(p\) and \(q\) for which \(l_1\) and \(l_2\) are parallel are respectively :

• A. \(-1, 3\)
• B. \(3, 1\)
• C. \(-3, -1\)
• D. \(-1, -3\)

Hint:

For parallel lines, one vector is a scalar multiple of the other. You can often solve these by inspection: since \(12 = 3 \times 4\), just multiply the second set by \(3\) to match the first, or divide the first set by \(3\) to find the second.

Answer 1

The Correct Option is C

Concept:
• Two lines with direction ratios \(\langle a_1, b_1, c_1 \rangle\) and \(\langle a_2, b_2, c_2 \rangle\) are parallel if their corresponding direction ratios are proportional.
• Condition for parallelism: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k\), where \(k\) is a constant.

Step 1: Set up the proportionality equation
Given direction ratios: Line \(l_1\): \(\langle 12, -3, 9 \rangle\) Line \(l_2\): \(\langle 4, q, -p \rangle\) Since they are parallel: \[ \frac{12}{4} = \frac{-3}{q} = \frac{9}{-p} \]

Step 2: Solve for \(q\)
Equate the first two ratios: \[ 3 = \frac{-3}{q} \implies q = \frac{-3}{3} = -1 \]

Step 3: Solve for \(p\)
Equate the first and third ratios: \[ 3 = \frac{9}{-p} \implies -p = \frac{9}{3} = 3 \implies p = -3 \]

Step 4: Identify the values
We have \(p = -3\) and \(q = -1\).