Question

If $\theta $ is the angle between the line $\frac{x + 1}{1}=\frac{y - 1}{2}=\frac{z - 2}{2}$ and the plane $2x-y+\sqrt{\lambda }z+4=0$ such that $sin\theta =\frac{1}{3}$ , then $\lambda $ is equal to

• A. $\frac{5}{3}$
• B. $\frac{5}{2}$
• C. $\frac{5}{4}$
• D. $\frac{3}{2}$

Answer 1

Answer: (A)

Here, the angle between the line and normal to the plane is
$cos\left(90 ^\circ - \theta \right)=\frac{2 - 2 + 2 \sqrt{\lambda }}{3 \left(\right. \sqrt{5 + \lambda } \left.\right)}$
$sin\theta =\frac{2 - 2 + 2 \sqrt{\lambda }}{3 \left(\right. \sqrt{5 + \lambda } \left.\right)}$
$\Rightarrow \frac{1}{3}=\frac{2 \sqrt{\lambda }}{3 \left(\right. \sqrt{5 + \lambda } \left.\right)}$
$\Rightarrow \lambda =5/3$