The locus represented by `xy+yz=0` is

Options

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

Answer

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

### Step 1: Rewrite the Equation
Start with the original equation:
\[ xy + yz = 0 \]
We can factor out \( y \) from the equation:
\[ y(x + z) = 0 \]

### Step 2: Analyze the Factors
From the factored equation \( y(x + z) = 0 \), we can set each factor to zero:
1. \( y = 0 \)
2. \( x + z = 0 \)

### Step 3: Interpret Each Factor
- The equation \( y = 0 \) represents the \( xz \)-plane in three-dimensional space, where the \( y \)-coordinate is always zero.
- 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.

### Step 4: Determine the Relationship Between the Planes
The two equations \( y = 0 \) and \( z = -x \) represent two different planes in three-dimensional space. To check if these planes are perpendicular, we can find their normal vectors:
- The normal vector for the plane \( y = 0 \) is \( \mathbf{n_1} = (0, 1, 0) \).
- The normal vector for the plane \( z = -x \) is \( \mathbf{n_2} = (1, 0, 1) \).

### Step 5: Check for Perpendicularity
To check if the planes are perpendicular, we compute the dot product of the normal vectors:
\[ \mathbf{n_1} \cdot \mathbf{n_2} = (0)(1) + (1)(0) + (0)(1) = 0 \]
Since the dot product is zero, the planes are perpendicular.

### Conclusion
The locus represented by the equation \( xy + yz = 0 \) 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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