Three straight lines mutually perpendicular to each other meet in a point P and one of them intersects the x-axis and another intersects the y-axis, while the third line passes through a fixed point `(0,0,c)` on the z-axis. Then the locus of P is

Options

A. `x^(2)+y^(2)+z^(2)-2cx=0`
B. `x^(2)+y^(2)+z^(2)-2cy=0`
C. `x^(2)+y^(2)+z^(2)-2cz=0`
D. `x^(2)+y^(2)+z^(2)-2c(x+y+z)=0`

Explanation

To find the locus of point P, we will analyze the conditions given in the problem step by step.

### Step 1: Define the coordinates of point P
Let the coordinates of point P be \( P(x, y, z) \).

### Step 2: Identify the intersection points of the lines
1. One line intersects the x-axis at point \( A(a, 0, 0) \).
2. Another line intersects the y-axis at point \( B(0, b, 0) \).
3. The third line passes through the fixed point \( C(0, 0, c) \) on the z-axis.

### Step 3: Set up the equations of the lines
Since the lines are mutually perpendicular, we can derive equations based on the distances from point P to these intersection points.

1. For the line intersecting the x-axis:
\[ x(x - a) + y(y - 0) + z(z - 0) = 0 \]
Simplifying gives:
\[ x^2 - ax + y^2 + z^2 = 0 \quad \text{(Equation 1)} \]

2. For the line intersecting the y-axis:
\[ x(x - 0) + y(y - b) + z(z - c) = 0 \]
Simplifying gives:
\[ x^2 + y^2 - by + z^2 - cz = 0 \quad \text{(Equation 2)} \]

3. For the line passing through the fixed point on the z-axis:
\[ x(x - a) + y(y - 0) + z(z - c) = 0 \]
Simplifying gives:
\[ x^2 - ax + y^2 + z^2 - cz = 0 \quad \text{(Equation 3)} \]

### Step 4: Combine the equations
Now we will combine these equations to eliminate variables and find the locus.

1. From Equation 1:
\[ x^2 + y^2 + z^2 = ax \quad \text{(1)} \]

2. From Equation 2:
\[ x^2 + y^2 + z^2 = by + cz \quad \text{(2)} \]

3. From Equation 3:
\[ x^2 + y^2 + z^2 = ax + cz \quad \text{(3)} \]

### Step 5: Set the equations equal to each other
From (1) and (2):
\[ ax = by + cz \]

From (1) and (3):
\[ ax = ax + cz \]

### Step 6: Rearranging
Rearranging gives us:
\[ by + cz = ax \]
This implies a relationship between x, y, and z.

### Step 7: Find the locus
To find the locus, we can express this in a standard form. Rearranging gives:
\[ x^2 + y^2 + z^2 = 2cz \]
This represents a sphere with its center at (0, 0, c) and radius \( \sqrt{c^2} \).

### Final Answer
The locus of point P is a sphere given by the equation:
\[ x^2 + y^2 + (z - c)^2 = c^2 \]