What is the equation of the set of points that are 5 units from point (2,3,4) ?

Options

A. `2x+3y+4z=5`
B. `x^(2)+y^(2)+z^(2)-4x-6y-8z=25`
C. `(x-2)^(2) +(y-3)^(2)+(z-4)^(2)=25`
D. `x^(2)+y^(2)+z^(2)=5`

Explanation

To find the equation of the set of points that are 5 units away from the point (2, 3, 4), we can use the formula for a sphere in three-dimensional space. The general equation for a sphere with center (a, b, c) and radius r is given by:

\[ (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 \]

### Step-by-Step Solution:

1. **Identify the center and radius**:
- The center of the sphere is given as the point (2, 3, 4).
- The radius is given as 5 units.

2. **Substitute the center coordinates into the equation**:
- Here, \( a = 2 \), \( b = 3 \), and \( c = 4 \).

3. **Substitute the radius into the equation**:
- The radius \( r = 5 \), so \( r^2 = 5^2 = 25 \).

4. **Write the equation of the sphere**:
- Substituting the values into the sphere equation:
\[ (x - 2)^2 + (y - 3)^2 + (z - 4)^2 = 25 \]

5. **Final equation**:
- The equation of the set of points that are 5 units from the point (2, 3, 4) is:
\[ (x - 2)^2 + (y - 3)^2 + (z - 4)^2 = 25 \]