What is the equation off the sphere with unit radius having center at the origin?

Options

A. `x^2+y^2+z^2=0`
B. `x^2+y^2+z^2=1`
C. `x^2+y^2+z^2=0`
D. `x^2+y^2+z^2=3`

Answer

To find the equation of a sphere with a unit radius centered at the origin, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the general equation of a sphere**:
The equation of a sphere with center at \((a, b, c)\) and radius \(r\) is given by:
\[ (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 \]

2. **Identify the center and radius**:
In this case, the center of the sphere is at the origin, which means \(a = 0\), \(b = 0\), and \(c = 0\). The radius is given as a unit radius, so \(r = 1\).

3. **Substitute the values into the equation**:
Plugging in the values of \(a\), \(b\), \(c\), and \(r\) into the general equation:
\[ (x - 0)^2 + (y - 0)^2 + (z - 0)^2 = 1^2 \]
This simplifies to:
\[ x^2 + y^2 + z^2 = 1 \]

4. **Final equation**:
Therefore, the equation of the sphere with unit radius centered at the origin is:
\[ x^2 + y^2 + z^2 = 1 \]