Question

If the angle θ between the line x+11=y−12=z−22 and the plane 2x−y+λz+4=0 is such that sin⁡θ=13, then value of λ is

• A. (A) $\frac{ - 3}{5}$
• B. (B) $\frac{5}{3}$
• C. (C) $\frac{ - 4}{3}$
• D. (D ) $\frac{ 3}{4}$

Answer 1

Answer: (B)

(B) $\frac{5}{3}$

Answer 2

Explanation:
Given:Equation of line, x+11=y−12=z−22…(i)and equation of the plane, 2x−y+λz+4=0…(ii)We have to find value of λ such that the angle θ between line and plane is given by sin⁡θ=13The angle θ between the line (i) and plane (ii) is given by sin⁡θ=|1.2+2⋅(−1)+2⋅(λ)1+4+4⋅4+1+λ|[Using formula of Angle Between a Line and a Plane] ⇒13=|2λ3⋅4+1+λ|[sin⁡θ=13( Given )]⇒5+λ=2λ⇒5+λ=4λ [Squaring both sides] ⇒3λ=5⇒λ=53Hence, the correct option is (B).