1. Key Concepts and Formulas

   • General Equation of a Sphere: The equation of a sphere is given by $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$. The center of this sphere is $C = (-u, -v, -w)$.
   • Diameter Property: The center of a sphere is the midpoint of any of its diameters.
   • Midpoint Formula in 3D: If $P_1 = (x_1, y_1, z_1)$ and $P_2 = (x_2, y_2, z_2)$ are the endpoints of a line segment, its midpoint $M = (x_m, y_m, z_m)$ is given by $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$.

2. Step-by-Step Solution

   Step 1: Determine the Center of the Sphere We are given the equation of the sphere: $x^2 + y^2 + z^2 - 6x - 12y - 2z + 20 = 0$ To find the center of the sphere, we compare this equation with the general form: $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$ By comparing the coefficients of $x, y,$ and $z$ from the given equation with the general form:

   • Coefficient of $x$: $2u = -6 \Rightarrow u = -3$
   • Coefficient of $y$: $2v = -12 \Rightarrow v = -6$
   • Coefficient of $z$: $2w = -2 \Rightarrow w = -1$ The center of the sphere, $C = (-u, -v, -w)$, is therefore: $C = (-(-3), -(-6), -(-1)) = (3, 6, 1)$ This center point is crucial because it acts as the midpoint of any diameter.

   Step 2: Use the Midpoint Formula to Find the Other End of the Diameter We know that the center of the sphere is the midpoint of any diameter. Let $P_1 = (x_1, y_1, z_1) = (2,3,5)$ be the given end of the diameter, and let $P_2 = (x_2, y_2, z_2)$ be the coordinates of the other end we need to find. The center of the sphere is $C = (x_c, y_c, z_c) = (3,6,1)$. Using the midpoint formula for 3D coordinates: $(x_c, y_c, z_c) = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$ Substitute the known values: $(3, 6, 1) = \left(\frac{2+x_2}{2}, \frac{3+y_2}{2}, \frac{5+z_2}{2}\right)$ Now, we equate the corresponding coordinates to solve for $x_2, y_2,$ and $z_2$:

   • For the x-coordinate: $3 = \frac{2+x_2}{2} \Rightarrow 6 = 2+x_2 \Rightarrow x_2 = 4$
   • For the y-coordinate: $6 = \frac{3+y_2}{2} \Rightarrow 12 = 3+y_2 \Rightarrow y_2 = 9$
   • For the z-coordinate: $1 = \frac{5+z_2}{2} \Rightarrow 2 = 5+z_2 \Rightarrow z_2 = -3$ Thus, the coordinates of the other end of the diameter are $P_2 = (4, 9, -3)$.

3. Common Mistakes & Tips

   • Sign Errors in Center Calculation: A frequent mistake is failing to correctly apply the negative signs when determining the center coordinates from the general equation (e.g., using $u, v, w$ instead of $-u, -v, -w$). Always remember that the center is $(-u, -v, -w)$.
   • Careless Algebraic Manipulation: When solving for the unknown coordinates using the midpoint formula, ensure careful algebraic steps to avoid errors in addition, subtraction, or multiplication.
   • Verification (Optional but Recommended): You can quickly check if the given point $(2,3,5)$ actually lies on the sphere by substituting its coordinates into the sphere's equation. This confirms the validity of the starting point. ($2^2+3^2+5^2 - 6(2) - 12(3) - 2(5) + 20 = 4+9+25-12-36-10+20 = 38-58+20 = 0$).

4. Summary To find the other end of a diameter, we first identified the center of the sphere by comparing its given equation to the standard general form. The center of the sphere is a crucial point as it serves as the midpoint for any diameter. We then applied the 3D midpoint formula, using the known center and one endpoint of the diameter, to algebraically determine the coordinates of the unknown other endpoint. This systematic approach ensures an accurate solution.

5. Final Answer The coordinates of the other end of the diameter are $(4, 9, -3).$ which corresponds to option (C).

The final answer is $\boxed{(4, 9, -3)}$.