The value of \(k\) for which the lines \(\displaystyle\frac{2x-3}{4}=\frac{3-y}{k}=\frac{z-2}{-2}\) and \(\displaystyle\frac{x-2}{1}=\frac{y}{4}=\frac{5-z}{3}\) are perpendicular to each other is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
Direction ratios are read only from the standard form \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\), where x, y and z each have coefficient +1.
First line: divide the first fraction by 2, and write \(3 - y = -(y - 3)\):
\( \displaystyle \frac{x - \frac{3}{2}}{2} = \frac{y - 3}{-k} = \frac{z - 2}{-2} \)
Direction ratios: \((2, -k, -2)\).
Second line: write \(5 - z = -(z - 5)\):
\( \displaystyle \frac{x - 2}{1} = \frac{y - 0}{4} = \frac{z - 5}{-3} \)
Direction ratios: \((1, 4, -3)\).
The lines are perpendicular when \(a_1a_2 + b_1b_2 + c_1c_2 = 0\):
\(2(1) + (-k)(4) + (-2)(-3) = 0\)
\(8 - 4k = 0\)
\(k = 2\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{5}{2}\) — Incorrect. This comes from taking 4 as the first direction ratio without dividing \(2x - 3\) by 2: \(4 - 4k + 6 = 0\) gives \(k = \frac{5}{2}\).
Option (2) \(-\displaystyle\frac{5}{2}\) — Incorrect. This comes from both slips: first ratio 4 and second ratio k instead of −k, so \(4 + 4k + 6 = 0\).
Option (4) −2 — Incorrect. This comes from taking k instead of −k as the second ratio (ignoring the sign of \(3 - y\)): \(2 + 4k + 6 = 0\) gives \(k = -2\).