A line segment joining (1,0,1) and the origin (0,0,0) is resolved about the x-axis to form a right circular cone. If (x,y,z) is any point on the cone, other than the origin, then it satisfies the equation

Options

A. `x^(2)-2y^(2)-z^(2)=0`
B. `x^(2)-y^(2)-z^(2)=0`
C. `2x^(2)-y^(2)-2z^(2)=0`
D. `x^(2)-2y^(2)-2z^(2)=0`

Explanation

To solve the problem, we need to find the equation that any point \((x, y, z)\) on the cone formed by rotating the line segment joining the points \((1, 0, 1)\) and the origin \((0, 0, 0)\) about the x-axis satisfies.

### Step-by-Step Solution:

1. **Identify the Points**:
The line segment connects the points \((1, 0, 1)\) and \((0, 0, 0)\).

2. **Understand the Geometry**:
When we rotate the line segment about the x-axis, it forms a right circular cone. The apex of the cone is at the origin \((0, 0, 0)\) and the point \((1, 0, 1)\) lies on the surface of the cone.

3. **Calculate the Lengths**:
- The distance from the origin \((0, 0, 0)\) to the point \((1, 0, 1)\) is given by:
\[ OP = \sqrt{(1-0)^2 + (0-0)^2 + (1-0)^2} = \sqrt{1 + 0 + 1} = \sqrt{2} \]
- The distance from the point \((x, y, z)\) on the cone to the origin is:
\[ ON = \sqrt{x^2 + y^2 + z^2} \]
- The distance from the point \((x, 0, 0)\) on the x-axis (the projection of point \((x, y, z)\) on the x-axis) to the point \((1, 0, 1)\) is:
\[ OC = \sqrt{(1-x)^2 + (0-0)^2 + (1-0)^2} = \sqrt{(1-x)^2 + 1} \]

4. **Set Up the Proportionality**:
From the geometry of the cone, we have:
\[ \frac{OP}{OM} = \frac{ON}{OC} \]
Substituting the distances we calculated:
\[ \frac{\sqrt{x^2 + y^2 + z^2}}{\sqrt{2}} = \frac{\sqrt{x^2 + y^2 + z^2}}{\sqrt{(1-x)^2 + 1}} \]

5. **Cross Multiply**:
Cross multiplying gives us:
\[ \sqrt{x^2 + y^2 + z^2} \cdot \sqrt{(1-x)^2 + 1} = \sqrt{2} \cdot \sqrt{x^2 + y^2 + z^2} \]

6. **Square Both Sides**:
Squaring both sides leads to:
\[ (x^2 + y^2 + z^2)((1-x)^2 + 1) = 2(x^2 + y^2 + z^2) \]

7. **Simplify**:
Expanding the left side:
\[ (x^2 + y^2 + z^2)((1 - 2x + x^2) + 1) = (x^2 + y^2 + z^2)(2 - 2x + x^2) \]
This simplifies to:
\[ 2x^2 + 2y^2 + 2z^2 - 2x^3 - 2xy^2 - 2xz^2 = 2x^2 + 2y^2 + 2z^2 \]

8. **Rearranging**:
Rearranging gives us:
\[ 2x^2 = x^2 + y^2 + z^2 \]
Thus:
\[ x^2 - y^2 - z^2 = 0 \]

9. **Final Equation**:
Therefore, the equation that any point \((x, y, z)\) on the cone satisfies is:
\[ x^2 - y^2 - z^2 = 0 \]