If the line x = y = z intersect the line x sin A + y sin B + z sin C = 2d^2 , x sin 2A + y sin 2B + z sin 2C = d^2 then is equal to

Q: If the line x = y = z intersect the line x sin A + y sin B + z sin C = 2d², x sin 2A + y sin 2B + z sin 2C = d² then $sin\frac{A}{2} sin\frac{B}{2} sin\frac{C}{2}$ is equal to (where A + B + C = π)

(A) $\frac{1}{16}$

(B) $\frac{1}{8}$

(C) $\frac{1}{32}$

(D) $\frac{1}{12}$

Sol. Let (λ, λ, λ) be the point of intersection of two lines.

⇒ λ(sin A + sin B + sin C) = 2d² and λ(sin 2A + sin 2B + sin 2C) = d²

$\large \frac{sin 2A + sin 2B + sin 2C}{sin A + sin B + sin C} = \frac{1}{2}$

$\large sin\frac{A}{2} sin\frac{B}{2} sin\frac{C}{2} = \frac{1}{16} $