The locus represented by `xy+yz=0` is a pair of

Options

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

Explanation

To solve the problem, we need to analyze the equation given:

**Step 1: Start with the given equation.**
The equation we have is:
\[ xy + yz = 0 \]

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

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

**Step 4: Analyze the cases.**
- The first case \( y = 0 \) represents the \( xz \)-plane, which is a plane in 3D space where the y-coordinate is always zero.
- The second case \( x + z = 0 \) can be rewritten as \( z = -x \), which represents another plane in 3D space.

**Step 5: Conclusion about the locus.**
The locus represented by the equation \( xy + yz = 0 \) consists of two planes:
1. The plane where \( y = 0 \) (the \( xz \)-plane).
2. The plane where \( z = -x \).

Since these two planes intersect along the line where \( y = 0 \) and \( z = -x \), they are not parallel and they are not perpendicular lines either. However, they do represent two distinct planes in 3D space.

Thus, the correct answer is that the locus represented by the equation is a pair of perpendicular planes.

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

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