If `(2, 3, 5)` is one end of a diameter of the sphere `x^(2)+y^(2)+z^(2)-6x-12y-2z+20=0`, then the coordinates of the other end of the diameter are

Options

A. `(4, 9, -3)`
B. `(4, -3, 3)`
C. `(4, 3, 5)`
D. `(4, 3, -3)`

Answer

To find the coordinates of the other end of the diameter of the sphere given one end as (2, 3, 5), we first need to determine the center of the sphere from its equation.

### Step-by-Step Solution:

1. **Write the equation of the sphere**:
The equation given is:
\[ x^2 + y^2 + z^2 - 6x - 12y - 2z + 20 = 0 \]

2. **Rearrange the equation**:
We can rearrange the equation to identify the center of the sphere by completing the square for each variable (x, y, z).

3. **Complete the square for x**:
\[ x^2 - 6x \rightarrow (x - 3)^2 - 9 \]

4. **Complete the square for y**:
\[ y^2 - 12y \rightarrow (y - 6)^2 - 36 \]

5. **Complete the square for z**:
\[ z^2 - 2z \rightarrow (z - 1)^2 - 1 \]

6. **Substitute back into the equation**:
Substitute the completed squares back into the equation:
\[ (x - 3)^2 - 9 + (y - 6)^2 - 36 + (z - 1)^2 - 1 + 20 = 0 \]
Simplifying this gives:
\[ (x - 3)^2 + (y - 6)^2 + (z - 1)^2 - 26 = 0 \]
Thus, we have:
\[ (x - 3)^2 + (y - 6)^2 + (z - 1)^2 = 26 \]

7. **Identify the center of the sphere**:
The center of the sphere is at the point (3, 6, 1).

8. **Use the midpoint formula**:
If (2, 3, 5) is one end of the diameter and (a, b, c) is the other end, the center (3, 6, 1) is the midpoint:
\[ \left(\frac{2 + a}{2}, \frac{3 + b}{2}, \frac{5 + c}{2}\right) = (3, 6, 1) \]

9. **Set up equations for each coordinate**:
From the midpoint formula, we can set up the following equations:
\[ \frac{2 + a}{2} = 3 \quad \Rightarrow \quad 2 + a = 6 \quad \Rightarrow \quad a = 4 \]
\[ \frac{3 + b}{2} = 6 \quad \Rightarrow \quad 3 + b = 12 \quad \Rightarrow \quad b = 9 \]
\[ \frac{5 + c}{2} = 1 \quad \Rightarrow \quad 5 + c = 2 \quad \Rightarrow \quad c = -3 \]

10. **Conclusion**:
Therefore, the coordinates of the other end of the diameter are:
\[ (4, 9, -3) \]