The center of the sphere which passes through `(a,0,0),(0,b,0),(0,0,c) and (0,0,0)` is

Options

A. `(a/2,0,0)`
B. `(0,b/2,0)`
C. `(0,0,c/2)`
D. `(a/2,b/2,c/2)`

Answer

To find the center of the sphere that passes through the points \( (a, 0, 0) \), \( (0, b, 0) \), \( (0, 0, c) \), and the origin \( (0, 0, 0) \), we can follow these steps:

### Step 1: Define the center of the sphere
Let the center of the sphere be denoted as \( H(x, y, z) \).

### Step 2: Set up the distance equations
Since the sphere passes through all four points, the distances from the center \( H \) to each of these points must be equal. Therefore, we can set up the following equations based on the distance formula:

1. Distance from \( H \) to the origin \( O(0, 0, 0) \):
\[ HO = \sqrt{x^2 + y^2 + z^2} \]

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

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

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

### Step 3: Equate distances
Since all distances are equal, we can set the equations as follows:

1. From \( HO = HA \):
\[ \sqrt{x^2 + y^2 + z^2} = \sqrt{(x - a)^2 + y^2 + z^2} \]

Squaring both sides gives:
\[ x^2 + y^2 + z^2 = (x - a)^2 + y^2 + z^2 \]
Simplifying this, we get:
\[ x^2 = x^2 - 2ax + a^2 \implies 2ax = a^2 \implies x = \frac{a}{2} \]

2. From \( HO = HB \):
\[ \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (y - b)^2 + z^2} \]

Squaring both sides gives:
\[ x^2 + y^2 + z^2 = x^2 + (y - b)^2 + z^2 \]
Simplifying this, we get:
\[ y^2 = y^2 - 2by + b^2 \implies 2by = b^2 \implies y = \frac{b}{2} \]

3. From \( HO = HC \):
\[ \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + y^2 + (z - c)^2} \]

Squaring both sides gives:
\[ x^2 + y^2 + z^2 = x^2 + y^2 + (z - c)^2 \]
Simplifying this, we get:
\[ z^2 = z^2 - 2cz + c^2 \implies 2cz = c^2 \implies z = \frac{c}{2} \]

### Step 4: Conclusion
Thus, the coordinates of the center \( H \) of the sphere are:
\[ H\left(\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right) \]

### Final Answer
The center of the sphere is \( \left(\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right) \).

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