3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
The planes $x + y - z = 0, y + z - x = 0, z + x - y = 0$ meet :
in a line
taken two at a time in parallel lines
in a unique point
None of these
Step-by-Step Solution
Key Concept: A non-zero coefficient determinant guarantees a unique solution for three planes in 3D space.
The coefficient determinant of the system is $\begin{vmatrix} 1 & 1 & -1 \\ 1 & -1 & -1 \\ 1 & -1 & 1 \end{vmatrix} \neq 0$, which equals $-4 \neq 0$. Since the determinant is non-zero, the system has a unique solution and the three planes meet at exactly one point.
Correct Answer: 3