Find the equation of the locus of a point whose distanace from the z-axis is equal to its distance from the 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 1: Understand the Distances
The distance of a point \( A(x, y, z) \) from the z-axis can be determined by the formula for the distance from a point to a line. The z-axis can be represented by the points \( (0, 0, z) \). The distance \( p \) from point \( A \) to the z-axis is given by:
\[ p = \sqrt{x^2 + y^2} \]

The distance of point \( A \) from the xy-plane (which is represented by the plane \( z = 0 \)) is simply the absolute value of the z-coordinate:
\[ q = |z| \]

### Step 2: Set the Distances Equal
According to the problem, the distance from the z-axis is equal to the distance from the xy-plane. Therefore, we can set the two distances equal to each other:
\[ \sqrt{x^2 + y^2} = |z| \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ x^2 + y^2 = z^2 \]

### Step 4: Rearrange the Equation
The equation \( x^2 + y^2 = z^2 \) can be rearranged to express the locus in a standard form:
\[ x^2 + y^2 - z^2 = 0 \]

### Conclusion
The equation of the locus of points whose distance from the z-axis is equal to their distance from the xy-plane is:
\[ x^2 + y^2 - z^2 = 0 \]

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