3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
If the line $x = y = z$ intersect the line in $A x + \sin B y + \sin C z = 2d^2$, sin $2A x + \sin 2B y + \sin 2C z = d^2$ then $\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ is equal to : (where $A + B + C = \pi$)
$1/16$
$1/8$
$1/32$
$1/12$
Step-by-Step Solution
Key Concept: The intersection point of two parametric lines leads to a ratio condition that can be simplified using trigonometric identities.
Let $(\lambda, \lambda, \lambda)$ be the point of intersection of two lines. From the constraint equations $\lambda(\sin A + \sin B + \sin C) = 2d^2$ and $\lambda(\sin 2A + \sin 2B + \sin 2C) = d^2$, dividing gives $\frac{\sin 2A + \sin 2B + \sin 2C}{\sin A + \sin B + \sin C} = \frac{1}{2}$. Using product-to-sum formulas, this simplifies to $\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} = \frac{1}{16}$.
Correct Answer: 1