3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade None
Question:
A ray $M$ is sent along the line $\frac{x - 0}{2} = \frac{y - 2}{2} = \frac{z - 1}{0}$ and is reflected by the plane $x = 0$ at point $A$. The reflected ray is again reflected by the plane $x + 2y = 0$ at point $B$. The initial ray and final reflected ray meets at point $J$. Then:
The co-ordinates of point $B$ is $(4, -2, 1)$
The co-ordinates of point $J$ is $(-3, -1, 1)$
The centroid of $\triangle ABJ$ is $(0, 0, 0)$
The co-ordinates of point $J$ is $(2, -1, 1)$
Step-by-Step Solution
Key Concept: Reflection across a coordinate plane is found by negating the corresponding coordinate while preserving others.
A point on the line is $(2\lambda, 2 + 2\lambda, 1)$. When $x = 0$, we get $\lambda = 0$, so $A(0, 2, 1)$. To find the reflection of any point $P$ on the line $x = 0$, let $Q$ be its reflection; then $AQ$ is the reflected ray. The reflection reverses the direction relative to the $yz$-plane.
Correct Answer: I need to work through this step-by-step to find which options are correct.
**Step 1: Find point A (reflection on plane x = 0)**
The initial ray has parametric form: $(2\lambda, 2 + 2\lambda, 1)$
At plane $x = 0$: $2\lambda = 0