If θ is the angle between the line (x + 1/1)=(y - 1/2)=(z - 2/2) and the plane 2x-y+√λ z+4=0 such that sinθ =(1/3) , then λ is equal to

Question

If θ is the angle between the line (x + 1/1)=(y - 1/2)=(z - 2/2) and the plane 2x-y+√λ z+4=0 such that sinθ =(1/3) , then λ is equal to (A) (5/3) (B) (5/2) (C) (5/4) (D) (3/2)

Tardigrade Admin · Accepted Answer

Here, the angle between the line and normal to the plane is

cos (9 0^{\circ} - θ) = \frac{2 - 2 + 2 λ}{3 ( 5 + λ )}

s in θ = \frac{2 - 2 + 2 λ}{3 ( 5 + λ )}

\Rightarrow \frac{1}{3} = \frac{2 λ}{3 ( 5 + λ )}

\Rightarrow λ = 5/3

Q. If $θ$ is the angle between the line $\frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2}$ and the plane $2 x - y + λ z + 4 = 0$ such that $s in θ = \frac{1}{3}$ , then $λ$ is equal to

$\frac{5}{3}$

$\frac{5}{2}$

$\frac{5}{4}$

$\frac{3}{2}$

Here, the angle between the line and normal to the plane is
$cos (9 0^{\circ} - θ) = \frac{2 - 2 + 2 λ}{3 ( 5 + λ )}$
$s in θ = \frac{2 - 2 + 2 λ}{3 ( 5 + λ )}$
$\Rightarrow \frac{1}{3} = \frac{2 λ}{3 ( 5 + λ )}$
$\Rightarrow λ = 5/3$

Q. If θ is the angle between the line 1x+1​=2y−1​=2z−2​ and the plane 2x−y+λ​z+4=0 such that sinθ=31​ , then λ is equal to

35​

25​

45​

23​