What is the equation to the sphere whose centre is at (-2, 3, 4) and radius is 6 units

Options

A. `x^2+y^2+z^2+4x-6y-8z=7`
B. `x^2+y^2+z^2+6x - 4y - 8z = 7`
C. `x^2+y^2+z^2+4x-6y-8z=4`
D. `x^2+y^2+z^2+4x+6y+8z=4`

Answer

To find the equation of a sphere with a given center and radius, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the center and radius of the sphere**:
- The center of the sphere is given as \( (-2, 3, 4) \).
- The radius \( r \) is given as \( 6 \) units.

2. **Write the general equation of a sphere**:
- The general equation of a sphere with center \( (h, k, l) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]

3. **Substitute the center and radius into the equation**:
- Here, \( h = -2 \), \( k = 3 \), \( l = 4 \), and \( r = 6 \).
- Plugging these values into the equation gives:
\[ (x - (-2))^2 + (y - 3)^2 + (z - 4)^2 = 6^2 \]
- This simplifies to:
\[ (x + 2)^2 + (y - 3)^2 + (z - 4)^2 = 36 \]

4. **Expand the equation**:
- Now, we will expand the left-hand side:
\[ (x + 2)^2 = x^2 + 4x + 4 \]
\[ (y - 3)^2 = y^2 - 6y + 9 \]
\[ (z - 4)^2 = z^2 - 8z + 16 \]
- Combining these, we have:
\[ x^2 + 4x + 4 + y^2 - 6y + 9 + z^2 - 8z + 16 = 36 \]

5. **Combine like terms**:
- Adding the constant terms:
\[ 4 + 9 + 16 = 29 \]
- Therefore, we can rewrite the equation as:
\[ x^2 + y^2 + z^2 + 4x - 6y - 8z + 29 = 36 \]

6. **Rearranging the equation**:
- Subtract \( 29 \) from both sides:
\[ x^2 + y^2 + z^2 + 4x - 6y - 8z = 36 - 29 \]
- This simplifies to:
\[ x^2 + y^2 + z^2 + 4x - 6y - 8z = 7 \]

### Final Equation:
The final equation of the sphere is:
\[ x^2 + y^2 + z^2 + 4x - 6y - 8z = 7 \]