If two lines \(\displaystyle\frac{x-3}{2}=\frac{y-4}{5}=\frac{z}{4}\) and \(\displaystyle\frac{x-4}{3}=\frac{y-5}{6}=\frac{1-z}{k}\), are coplanar, then k is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −5
In the second line, \(\frac{1-z}{k} = \frac{z-1}{-k}\), so it is written in the standard form
\( \displaystyle \frac{x-4}{3}=\frac{y-5}{6}=\frac{z-1}{-k} \)
The lines pass through (3, 4, 0) and (4, 5, 1) with directions (2, 5, 4) and (3, 6, −k).
Two lines are coplanar when
\( \displaystyle \begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1\\ a_1 & b_1 & c_1\\ a_2 & b_2 & c_2 \end{vmatrix} = 0 \)
Here
\( \displaystyle \begin{vmatrix} 1 & 1 & 1\\ 2 & 5 & 4\\ 3 & 6 & -k \end{vmatrix} = 0 \)
Expanding along the first row:
\(1(-5k - 24) - 1(-2k - 12) + 1(12 - 15) = 0\)
\(-5k - 24 + 2k + 12 - 3 = 0\)
\(-3k - 15 = 0\)
\(k = -5\)
Tip: Rewrite \(\frac{1-z}{k}\) as \(\frac{z-1}{-k}\) before reading the direction ratios. Taking k as the third direction ratio gives k = 5, which is Option (1).
Note: This topic was not in the CUET 2026 syllabus.