If the angle θ between the line (x+1)/1=(y-1)/2=(z-2)/2 and the plane 2 x-y+√(λ) z+4= 0 is such that sin θ=1/3 the value of λ is

Correct answer: (A) 5/3. Angle between line and normal to plane is cos ((π)/2-θ )=(2-2+2 √(λ))/(3 × √(5+λ)) where θ is angle between line \& plane ⇒ sin θ=(2 √(λ))/(3 √(5+λ))=1/3 ⇒ λ=5/3 .

Q120 / 225 · JEE Main 2005 · Mathematics

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

Correct answer
A
$\frac{5}{3}$
B
$\frac{-3}{5}$
C
$\frac{3}{4}$
D
$\frac{-4}{3}$

Solution

Angle between line and normal to plane is $\cos \left(\frac{\pi}{2}-\theta\right)=\frac{2-2+2 \sqrt{\lambda}}{3 \times \sqrt{5+\lambda}}$ where $\theta$ is angle between line \& plane $$ \Rightarrow \sin \theta=\frac{2 \sqrt{\lambda}}{3 \sqrt{5+\lambda}}=\frac{1}{3} $$ $\Rightarrow \lambda=\frac{5}{3}$.

Hence, the correct answer is option: 'A'.

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