If the angle between the line $x=\frac{y-1}{2}=\frac{z-3}{\lambda}$ and the plane $x+2 y+3 z=4$ is $\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)$, then $\lambda$ equals

JEE Main · 2011 · 10 Apr 2011 · Offline · Three Dimensional Geometry

If the angle between the line

x = \frac{y - 1}{2} = \frac{z - 3}{λ}

and the plane

x + 2 y + 3 z = 4

cos^{- 1} (\frac{5}{14})

, then

λ

equals

(A) $\frac{3}{2}$
(B) $\frac{2}{5}$
(C) $\frac{5}{3}$
(D) $\frac{2}{3}$ ✓ Correct

Answer: (D)

Solution

The line has direction

(1, 2, λ)

and the plane normal is

(1, 2, 3)

. Given

cos θ = 5/14

for the angle between line and plane,

sin θ = 1 - \frac{5}{14} = \frac{3}{14}

. The line-plane angle also satisfies

sin θ = \frac{∣1 + 4 + 3 λ ∣}{5 + λ ^{2} 14}

, so

\frac{3}{14} = \frac{5 + 3 λ}{5 + λ ^{2} 14}

. Then

3 5 + λ^{2} = 5 + 3 λ

; squaring,

9 (5 + λ^{2}) = (5 + 3 λ)^{2}

reduces to

45 = 25 + 30 λ

, so

λ = \frac{2}{3}

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option

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