A,B,C are three points on the axis of x, y and z respectively at distance a,b,c from the orgain O, then the co-ordinates of the point which is equidistant from A,B,C and O is

Options

A. `(a,b,c)`
B. `(a/2,b/2,c/2)`
C. `(a/3,b/3,c/3)`
D. None

Answer

To find the coordinates of the point \( P \) that is equidistant from points \( A \), \( B \), \( C \), and the origin \( O \), we can follow these steps:

### Step 1: Define the Points
The points \( A \), \( B \), and \( C \) are defined as follows:
- Point \( A \) is on the x-axis at a distance \( a \) from the origin, so its coordinates are \( (a, 0, 0) \).
- Point \( B \) is on the y-axis at a distance \( b \) from the origin, so its coordinates are \( (0, b, 0) \).
- Point \( C \) is on the z-axis at a distance \( c \) from the origin, so its coordinates are \( (0, 0, c) \).
- The origin \( O \) has coordinates \( (0, 0, 0) \).

### Step 2: Set Up the Equidistance Condition
We need to find the point \( P(x, y, z) \) such that:
\[ P A = P B = P C = P O \]
This means we need to find the distances from \( P \) to each of these points.

### Step 3: Calculate Distances
1. **Distance from \( P \) to \( O \)**:
\[ P O = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{x^2 + y^2 + z^2} \]

2. **Distance from \( P \) to \( A \)**:
\[ P A = \sqrt{(x - a)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{(x - a)^2 + y^2 + z^2} \]

3. **Distance from \( P \) to \( B \)**:
\[ P B = \sqrt{(x - 0)^2 + (y - b)^2 + (z - 0)^2} = \sqrt{x^2 + (y - b)^2 + z^2} \]

4. **Distance from \( P \) to \( C \)**:
\[ P C = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - c)^2} = \sqrt{x^2 + y^2 + (z - c)^2} \]

### Step 4: Set Up Equations
From the equidistance condition, we can set up the following equations:
1. \( P O^2 = P A^2 \)
\[ x^2 + y^2 + z^2 = (x - a)^2 + y^2 + z^2 \]
Simplifying gives:
\[ x^2 = x^2 - 2ax + a^2 \implies 0 = a^2 - 2ax \implies 2ax = a^2 \implies x = \frac{a}{2} \]

2. \( P O^2 = P B^2 \)
\[ x^2 + y^2 + z^2 = x^2 + (y - b)^2 + z^2 \]
Simplifying gives:
\[ y^2 = y^2 - 2by + b^2 \implies 0 = b^2 - 2by \implies 2by = b^2 \implies y = \frac{b}{2} \]

3. \( P O^2 = P C^2 \)
\[ x^2 + y^2 + z^2 = x^2 + y^2 + (z - c)^2 \]
Simplifying gives:
\[ z^2 = z^2 - 2cz + c^2 \implies 0 = c^2 - 2cz \implies 2cz = c^2 \implies z = \frac{c}{2} \]

### Step 5: Final Coordinates
Thus, the coordinates of the point \( P \) that is equidistant from points \( A \), \( B \), \( C \), and \( O \) are:
\[ P\left(\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right) \]