3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade None
Question:
Let $A$ be a point $(1, 2, 3)$ in the given reference system. Then locus of the point $P$ in the first octant satisfying the equation $d(O, P) = d(A, P)$ does not contain:
1. any of the coordinates axes
2. any of the coordinates planes
3. any plane parallel to coordinates axes
4. any plane parallel to coordinate planes
Step-by-Step Solution
Key Concept: The equidistant locus from two points under taxicab metric decomposes into multiple cases based on signs, each yielding geometric objects.
Given $d(O,P) = d(A,P)$ where $A = (1,2,3)$, we obtain $|x|+|y|+|z| = |x-1|+|y-2|+|z-3|$. By analyzing all 8 sign combinations, most cases yield contradictions or define coordinate planes and planes parallel to them, with the final case $x+y+z=3$ representing a part of an oblique plane.
Correct Answer: 3