Find the co ordinates of the point equidistant from the points : `(2,0,0),(0,3,0),(0,0,8) `and `(0,0,0)`

Answer

To find the coordinates of the point equidistant from the points \( A(2,0,0) \), \( B(0,3,0) \), \( C(0,0,8) \), and the origin \( O(0,0,0) \), we will denote the required point as \( P(x, y, z) \).

### Step 1: Set up the distance equations
Since point \( P \) is equidistant from all four points, we can set up the following equations based on the distance formula:

1. Distance from \( P \) to \( O \):
\[ OP = \sqrt{x^2 + y^2 + z^2} \]

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

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

4. Distance from \( P \) to \( C \):
\[ CP = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - 8)^2} \]

### Step 2: Equate distances
We will first equate the distance from \( P \) to \( O \) and \( P \) to \( A \):
\[ OP = AP \]
Squaring both sides gives:
\[ x^2 + y^2 + z^2 = (x - 2)^2 + y^2 + z^2 \]
Expanding the right side:
\[ x^2 + y^2 + z^2 = x^2 - 4x + 4 + y^2 + z^2 \]
Cancelling \( x^2 \), \( y^2 \), and \( z^2 \) from both sides:
\[ 0 = -4x + 4 \]
Thus,
\[ 4x = 4 \implies x = 1 \]

### Step 3: Equate distances for \( B \)
Next, we equate the distance from \( P \) to \( O \) and \( P \) to \( B \):
\[ OP = BP \]
Squaring both sides gives:
\[ x^2 + y^2 + z^2 = x^2 + (y - 3)^2 + z^2 \]
Expanding the right side:
\[ x^2 + y^2 + z^2 = x^2 + (y^2 - 6y + 9) + z^2 \]
Cancelling \( x^2 \), \( y^2 \), and \( z^2 \) from both sides:
\[ 0 = -6y + 9 \]
Thus,
\[ 6y = 9 \implies y = \frac{9}{6} = \frac{3}{2} \]

### Step 4: Equate distances for \( C \)
Finally, we equate the distance from \( P \) to \( O \) and \( P \) to \( C \):
\[ OP = CP \]
Squaring both sides gives:
\[ x^2 + y^2 + z^2 = x^2 + y^2 + (z - 8)^2 \]
Expanding the right side:
\[ x^2 + y^2 + z^2 = x^2 + y^2 + (z^2 - 16z + 64) \]
Cancelling \( x^2 \) and \( y^2 \) from both sides:
\[ 0 = -16z + 64 \]
Thus,
\[ 16z = 64 \implies z = \frac{64}{16} = 4 \]

### Conclusion
The coordinates of the point \( P \) that is equidistant from the points \( A(2,0,0) \), \( B(0,3,0) \), \( C(0,0,8) \), and the origin \( O(0,0,0) \) are:
\[ P(1, \frac{3}{2}, 4) \]