Which one of the following planes contains the z-axis?

Options

A. `x-z=0`
B. `z+y=0`
C. `3x+2y=0`
D. `3x+2z=0`

Explanation

To determine which plane contains the z-axis, we need to analyze the equations of the planes given in the options. A plane contains the z-axis if every point on the z-axis satisfies the equation of that plane.

### Step-by-Step Solution:

1. **Understanding the Z-axis**:
The z-axis consists of all points where x = 0 and y = 0, so the points on the z-axis can be represented as (0, 0, z) for any value of z.

2. **General Form of a Plane**:
A plane in 3D space can be represented by an equation of the form Ax + By + Cz + D = 0, where A, B, C, and D are constants.

3. **Substituting Points on the Z-axis**:
To check if a plane contains the z-axis, we substitute x = 0 and y = 0 into the equation of the plane and see if it holds true for all values of z.

4. **Analyzing the Options**:
Let's analyze each option:

- **Option A**: If the equation is x = 0, substituting gives us 0 = 0, which is true. However, this does not represent a plane but rather a line along the z-axis.

- **Option B**: If the equation is y = 0, substituting gives us 0 = 0, which is also true. Similar to option A, this does not represent a plane.

- **Option C**: If the equation is 3x + 2y = z, substituting x = 0 and y = 0 gives us 0 = z. This is only true for z = 0, meaning it does not contain the entire z-axis.

- **Option D**: If the equation is z = 0, substituting gives us z = 0. This represents the xy-plane, which does not contain the z-axis.

5. **Conclusion**:
After analyzing the options, we find that none of the options provided contain the z-axis in their entirety. However, if we consider the equation of the form Ax + By + D = 0, where C = 0, then the plane would be parallel to the z-axis and thus would contain it.

Therefore, the correct answer is that none of the given options represent a plane that contains the z-axis.