The locus represented by xy + yz = 0 is

Options

A. A pair of parallel lines
B. A pair of parallel lines
C. A pair of parallel planes
D. A pair of perpendicular planes

Answer

To find the locus represented by the equation \( xy + yz = 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ xy + yz = 0 \]

2. **Factor the equation:**
We can factor out \( y \) from the left-hand side:
\[ y(x + z) = 0 \]

3. **Set each factor to zero:**
This gives us two cases:
- Case 1: \( y = 0 \)
- Case 2: \( x + z = 0 \)

4. **Interpret the cases geometrically:**
- **Case 1:** The equation \( y = 0 \) represents the \( xz \)-plane in three-dimensional space.
- **Case 2:** The equation \( x + z = 0 \) can be rewritten as \( z = -x \), which represents a plane that passes through the origin and has a slope of -1 in the \( xz \)-plane.

5. **Combine the results:**
The locus represented by the equation \( xy + yz = 0 \) consists of the union of the \( xz \)-plane and the plane defined by \( x + z = 0 \).

6. **Conclusion:**
Therefore, the locus represented by the equation \( xy + yz = 0 \) is a pair of planes:
- The \( xz \)-plane
- The plane \( x + z = 0 \)

### Final Answer:
The locus represented by \( xy + yz = 0 \) is a pair of perpendicular planes.