3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
The foot of the perpendicular from the point $O(0, 0, 0)$ to the line of intersection of the planes $x + y + z = 4$ and $2x + y + 3z = 1$ is point $A$. Then the equation of line $OA$ is:
$\frac{x}{2} = \frac{y}{1} = \frac{z}{3}$
$\frac{x}{8} = \frac{y}{29} = \frac{z}{-13}$
$\frac{x}{-2} = \frac{y}{1} = \frac{z}{1}$
None of these
Step-by-Step Solution
Key Concept: A family of planes is parameterized, and perpendicularity conditions determine specific planes within the family.
The family of planes containing line $x + y + z - 4 + λ(2x + y + 3z - 11) = 0$ passes through origin when $λ = -4$, giving $7x + 3y + 11z = 0$. For the plane perpendicular to this family: $7(1 + 2λ₁) + 3(1 + λ₁) + 11(1 + 3λ₁) = 0$ yields $λ₁ = -21/50$. The perpendicular line $OA$ has direction ratios $(8, 29, -13)$.
Correct Answer: 2