If the lines \(\displaystyle\frac{1-x}{3} = \frac{3y-6}{k} = \frac{3-z}{-2}\) and \(\displaystyle\frac{1-x}{2k}=\frac{y-5}{3}=\frac{6-z}{5}\) are perpendicular to each other, then \(k\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{10}{7}\)
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:
\( \displaystyle \frac{1 - x}{3} = \frac{x - 1}{-3},\quad \frac{3y - 6}{k} = \frac{y - 2}{\frac{k}{3}},\quad \frac{3 - z}{-2} = \frac{z - 3}{2} \)
Direction ratios: \(\left(-3, \frac{k}{3}, 2\right)\)
Second line:
\( \displaystyle \frac{1 - x}{2k} = \frac{x - 1}{-2k},\quad \frac{y - 5}{3},\quad \frac{6 - z}{5} = \frac{z - 6}{-5} \)
Direction ratios: \((-2k, 3, -5)\)
The lines are perpendicular, so \(a_1a_2 + b_1b_2 + c_1c_2 = 0\):
\( \displaystyle (-3)(-2k) + \frac{k}{3}(3) + 2(-5) = 0 \)
\(6k + k - 10 = 0\)
\( \displaystyle k = \frac{10}{7} \)
Tip: Reading the denominators directly, \((3, k, -2)\) and \((2k, 3, 5)\), gives \(6k + 3k - 10 = 0\), that is, \(k = \frac{10}{9}\), Option (2). The signs and coefficients of x, y, z must be made +1 first.