3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
Locus of point $P$ if $d(O, P) = k$, where $k$ is a positive constant number, represents:
a sphere of radius k
a set of eight planes forming an octahedron
a set of eight planes forming hexagonal prism
an infinite cylinder of radius k
Step-by-Step Solution
Key Concept: The locus of constant taxicab distance from the origin forms 8 planar regions, one in each octant.
When $d(O,P) = k$, the equation becomes $|x|+|y|+|z| = k$. This represents a set of 8 planes since each of the three axes can have positive or negative coefficients, creating 8 octants with planes that make equal intercepts of length $k$ on all three axes.
Correct Answer: 2