3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade None
Question:
The equation of the plane bisecting the acute angle between the planes $2x - y + 2z + 3 = 0$ and $3x - 2y + 6z + 8 = 0$ is :
$23x - 13y + 32z + 45 = 0$
$5x - y - 4z = 3$
$5x - y - 4z + 45 = 0$
$23x - 13y + 32z + 3 = 0$
Step-by-Step Solution
Key Concept: The acute angle bisector between two planes is found by equating the normalized plane equations when the dot product of normal vectors is positive.
Given that $a_1d_1 + b_1b_2 + c_1c_2 = 6 + 2 + 12 > 0$, the acute angle bisector is found by equating $\frac{2x - y + 2z + 3}{3} = \frac{3x - 2y + 6z + 8}{7}$. Cross-multiplying and simplifying yields $23x - 13y + 32z + 45 = 0$.
Correct Answer: 1