List the elements of the sets in question 1.

Answer

To solve the problem of finding the coordinates of the other end of the diameter of the sphere given one end and the equation of the sphere, we can follow these steps:

### Step 1: Identify the given information
We are given:
- One end of the diameter: \( (2, 3, 5) \)
- The equation of the sphere: \( x^2 + y^2 + z^2 - 6x - 12y - 2z + 20 = 0 \)

### Step 2: Rewrite the equation of the sphere
We can rewrite the equation of the sphere in standard form by completing the square for \( x \), \( y \), and \( z \).

1. For \( x \):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
2. For \( y \):
\[ y^2 - 12y = (y - 6)^2 - 36 \]
3. For \( z \):
\[ z^2 - 2z = (z - 1)^2 - 1 \]

Putting it all together, we have:
\[ (x - 3)^2 - 9 + (y - 6)^2 - 36 + (z - 1)^2 - 1 + 20 = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y - 6)^2 + (z - 1)^2 = 26 \]
Thus, the center of the sphere is \( (3, 6, 1) \) and the radius is \( \sqrt{26} \).

### Step 3: Use the midpoint formula
Let the coordinates of the other end of the diameter be \( (a, b, c) \). The midpoint of the diameter (which is the center of the sphere) can be calculated using the midpoint formula:
\[ \left( \frac{2 + a}{2}, \frac{3 + b}{2}, \frac{5 + c}{2} \right) = (3, 6, 1) \]

### Step 4: Set up equations
From the midpoint formula, we can set up the following equations:
1. \( \frac{2 + a}{2} = 3 \)
2. \( \frac{3 + b}{2} = 6 \)
3. \( \frac{5 + c}{2} = 1 \)

### Step 5: Solve for \( a \), \( b \), and \( c \)
1. From \( \frac{2 + a}{2} = 3 \):
\[ 2 + a = 6 \implies a = 4 \]
2. From \( \frac{3 + b}{2} = 6 \):
\[ 3 + b = 12 \implies b = 9 \]
3. From \( \frac{5 + c}{2} = 1 \):
\[ 5 + c = 2 \implies c = -3 \]

### Step 6: Conclusion
The coordinates of the other end of the diameter are \( (4, 9, -3) \).

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