The value of \(p\) so that the lines \(\displaystyle\frac{x-1}{-3}=\frac{2y-2}{2p}=\frac{z-3}{2}\) and \(\displaystyle\frac{x-1}{-3p}=\frac{y-1}{4}=\frac{6-z}{5}\) are at right angles is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{10}{13}\)
Direction ratios can be 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: \(\frac{2y-2}{2p} = \frac{2(y-1)}{2p} = \frac{y-1}{p}\), so
\( \displaystyle \frac{x-1}{-3} = \frac{y-1}{p} = \frac{z-3}{2},\quad \text{direction ratios } (-3,\ p,\ 2) \)
Second line: \(\frac{6-z}{5} = \frac{z-6}{-5}\), so
\( \displaystyle \frac{x-1}{-3p} = \frac{y-1}{4} = \frac{z-6}{-5},\quad \text{direction ratios } (-3p,\ 4,\ -5) \)
The lines are perpendicular when \(a_1a_2 + b_1b_2 + c_1c_2 = 0\):
\((-3)(-3p) + (p)(4) + (2)(-5) = 0\)
\(9p + 4p - 10 = 0\)
\( \displaystyle 13p = 10,\quad p = \frac{10}{13} \)
Why the other options are incorrect
Option (1) \(-\displaystyle\frac{10}{17}\) — Incorrect. Taking (−3, 2p, 2) and (−3p, 4, 5) straight from the printed denominators gives 9p + 8p + 10 = 0, that is p = −10/17; neither line was first put in standard form.
Option (2) \(-\displaystyle\frac{10}{13}\) — Incorrect. Using (−3, p, 2) but (−3p, 4, 5), with the sign of 6 − z not changed, gives 13p + 10 = 0, that is p = −10/13.
Option (4) \(\displaystyle\frac{10}{17}\) — Incorrect. Using (−3, 2p, 2), with the factor 2 of 2y not removed, and (−3p, 4, −5) gives 17p − 10 = 0, that is p = 10/17.