Find the equation of locus of a point whose distance from z-axis is equal to its distance from xy-plane.

Answer

To find the equation of the locus of a point whose distance from the z-axis is equal to its distance from the xy-plane, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Point**:
Let the point be \( P(x, y, z) \).

2. **Distance from the z-axis**:
The distance of point \( P \) from the z-axis can be calculated using the formula:
\[ \text{Distance from z-axis} = \sqrt{x^2 + y^2} \]

3. **Distance from the xy-plane**:
The distance of point \( P \) from the xy-plane is given by:
\[ \text{Distance from xy-plane} = |z| \]

4. **Set the Distances Equal**:
According to the problem, the distance from the z-axis is equal to the distance from the xy-plane:
\[ \sqrt{x^2 + y^2} = |z| \]

5. **Square Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ x^2 + y^2 = z^2 \]

6. **Rearrange the Equation**:
Rearranging gives us:
\[ x^2 + y^2 - z^2 = 0 \]

7. **Final Equation of Locus**:
Thus, the equation of the locus of the point is:
\[ x^2 + y^2 - z^2 = 0 \]