Question

The equation $xy = 0$ in three-dimensional space represents:

• A. A pair of straight lines
• B. A plane
• C. A pair of planes at right angles
• D. A pair of parallel planes

Answer 1

The Correct Option is C

1. Analyze the equation \( xy = 0 \):

In three-dimensional space, \( xy = 0 \) implies either \( x = 0 \) or \( y = 0 \).

2. Interpret geometrically:

\( x = 0 \) represents the \( yz \)-plane, and \( y = 0 \) represents the \( xz \)-plane. These two planes intersect at the \( z \)-axis and are perpendicular to each other.

3. Match the result to the options:

The equation represents a pair of perpendicular planes, corresponding to option (C).

Correct Answer: (C) A pair of planes at right angles

Answer 2

The equation $ xy = 0 $ in three-dimensional space represents a pair of planes at right angles.

The equation $ xy = 0 $ implies that either $ x = 0 $ or $ y = 0 $ (or both).

• $ x = 0 $: This is the equation of a plane in 3D space; it's the $ yz $-plane.
• $ y = 0 $: This is also the equation of a plane in 3D space; it's the $ xz $-plane.

These two planes ($ yz $-plane and $ xz $-plane) intersect at a right angle along the $ z $-axis. 

Therefore, the correct answer is ${\text{(C)}} $.