Find the equation of the sphere passing through `(0, 0, 0), (1, 0, 0), (0, 1, 0) and (0, 0, 1)`.

Answer

To find the equation of the sphere passing through the points (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1), we can use the general equation of a sphere in three-dimensional space:

### Step 1: General Equation of a Sphere
The general equation of a sphere can be expressed as:
\[ x^2 + y^2 + z^2 + 2ax + 2by + 2cz + d = 0 \]
where \((a, b, c)\) is the center of the sphere and \(d\) is a constant.

### Step 2: Substitute the First Point (0, 0, 0)
Substituting the first point (0, 0, 0) into the equation:
\[ 0^2 + 0^2 + 0^2 + 2a(0) + 2b(0) + 2c(0) + d = 0 \]
This simplifies to:
\[ d = 0 \]

### Step 3: Substitute the Second Point (1, 0, 0)
Now, substitute the second point (1, 0, 0) into the equation:
\[ 1^2 + 0^2 + 0^2 + 2a(1) + 2b(0) + 2c(0) + 0 = 0 \]
This simplifies to:
\[ 1 + 2a = 0 \implies 2a = -1 \implies a = -\frac{1}{2} \]

### Step 4: Substitute the Third Point (0, 1, 0)
Next, substitute the third point (0, 1, 0):
\[ 0^2 + 1^2 + 0^2 + 2a(0) + 2b(1) + 2c(0) + 0 = 0 \]
This simplifies to:
\[ 1 + 2b = 0 \implies 2b = -1 \implies b = -\frac{1}{2} \]

### Step 5: Substitute the Fourth Point (0, 0, 1)
Finally, substitute the fourth point (0, 0, 1):
\[ 0^2 + 0^2 + 1^2 + 2a(0) + 2b(0) + 2c(1) + 0 = 0 \]
This simplifies to:
\[ 1 + 2c = 0 \implies 2c = -1 \implies c = -\frac{1}{2} \]

### Step 6: Write the Equation of the Sphere
Now we have \(a = -\frac{1}{2}\), \(b = -\frac{1}{2}\), \(c = -\frac{1}{2}\), and \(d = 0\). Substituting these values back into the general equation of the sphere gives:
\[ x^2 + y^2 + z^2 - x - y - z = 0 \]

### Step 7: Rearranging the Equation
Rearranging the equation, we get:
\[ x^2 - x + y^2 - y + z^2 - z = 0 \]

### Step 8: Completing the Square
To express this in standard form, we complete the square for each variable:
\[ \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} + \left(y - \frac{1}{2}\right)^2 - \frac{1}{4} + \left(z - \frac{1}{2}\right)^2 - \frac{1}{4} = 0 \]
This simplifies to:
\[ \left(x - \frac{1}{2}\right)^2 + \left(y - \frac{1}{2}\right)^2 + \left(z - \frac{1}{2}\right)^2 = \frac{3}{4} \]

### Final Equation of the Sphere
Thus, the equation of the sphere is:
\[ \left(x - \frac{1}{2}\right)^2 + \left(y - \frac{1}{2}\right)^2 + \left(z - \frac{1}{2}\right)^2 = \left(\frac{\sqrt{3}}{2}\right)^2 \]