Question

The number of distinct real values of \(\lambda\) for which the lines \(\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{\lambda^2}\) and \(\frac{x-3}{1}=\frac{y-2}{\lambda^2}=\frac{z-1}{2}\) are coplanar is :

• A. 4
• B. 1
• C. 2
• D. 3

Hint:

Two lines are coplanar when the scalar triple product of the join of two points and the two direction vectors is zero.

Answer 1

The Correct Option is D

Points: \((1,2,-3)\) and \((3,2,1)\); join \(=(2,0,4)\). Directions: \((1,2,\lambda^2)\) and \((1,\lambda^2,2)\).
Coplanarity: \(\begin{vmatrix}2&0&4\\1&2&\lambda^2\\1&\lambda^2&2\end{vmatrix}=0\).
\(2(4-\lambda^4)-0+4(\lambda^2-2)=0\Rightarrow-2\lambda^4+4\lambda^2=0\Rightarrow\lambda^2(\lambda^2-2)=0\).
\(\lambda=0,\ \sqrt2,\ -\sqrt2\): 3 distinct real values.
Answer: (4) 3