For every point `P(x ,\ y ,\ z)` on the xy-plane, a. `x=0` b. `y=0` c. `z=0` d. `x=y=z=0`

Options

A. `x=0`
B. `y=0`
C. `z=0`
D. none of these

Explanation

To determine the correct option for every point \( P(x, y, z) \) on the xy-plane, we need to analyze the characteristics of points that lie on this plane.

### Step-by-Step Solution:

1. **Understanding the xy-plane**:
- The xy-plane is defined as the plane where the z-coordinate is always zero. This means that any point on the xy-plane can be represented as \( P(x, y, 0) \).

2. **Identifying the coordinates**:
- For a point \( P \) on the xy-plane, the coordinates are \( (x, y, z) \) where \( z = 0 \). Therefore, the z-coordinate must be zero for any point on this plane.

3. **Analyzing the options**:
- a. \( x = 0 \): This is not true for all points on the xy-plane, as x can take any value.
- b. \( y = 0 \): This is also not true for all points on the xy-plane, as y can take any value.
- c. \( z = 0 \): This is true for all points on the xy-plane, as established earlier.
- d. \( x = y = z = 0 \): This is only true for the origin point (0, 0, 0) and not for all points on the xy-plane.

4. **Conclusion**:
- The correct option is **c. \( z = 0 \)**, as this condition must hold true for every point on the xy-plane.

### Final Answer:
The correct option is **c. \( z = 0 \)**.