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

JEE Main · 2005 · 08 May 2005 · Offline · Three Dimensional Geometry

If the angle

θ

between the line

\frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2}

and the plane

2 x - y + λ z + 4 = 0

is such that

sin θ = \frac{1}{3}

, the value of

λ

(A) $\frac{5}{3}$ ✓ Correct
(B) $\frac{- 3}{5}$
(C) $\frac{3}{4}$
(D) $\frac{- 4}{3}$

Answer: (A)

Solution

The line has direction ratios

(1, 2, 2)

and the plane has normal

(2, - 1, λ)

. The angle

θ

between a line and a plane satisfies

sin θ = \frac{∣ b \cdot n ∣}{∣ b ∣ ∣ n ∣} = \frac{∣2 - 2 + 2 λ ∣}{3 5 + λ} = \frac{2 λ}{3 5 + λ}

. Setting this equal to

\frac{1}{3}

gives

2 λ = 5 + λ

; squaring,

4 λ = 5 + λ

, so

λ = \frac{5}{3}

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option

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