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 at point \( A(2, 3, 5) \) and the equation of the sphere, we can follow these steps:

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

### Step 2: Rearrange the equation to find the center of the sphere
To find the center of the sphere, we need to rewrite the equation in the standard form of a sphere. We can do this by completing the square for \( x \), \( y \), and \( z \).

1. For \( x \):
\[ x^2 - 6x \quad \text{(complete the square)} \quad = (x - 3)^2 - 9 \]

2. For \( y \):
\[ y^2 - 12y \quad \text{(complete the square)} \quad = (y - 6)^2 - 36 \]

3. For \( z \):
\[ z^2 - 2z \quad \text{(complete the square)} \quad = (z - 1)^2 - 1 \]

Now substituting these back into the equation:
\[ (x - 3)^2 - 9 + (y - 6)^2 - 36 + (z - 1)^2 - 1 + 20 = 0 \]

Combine the constants:
\[ (x - 3)^2 + (y - 6)^2 + (z - 1)^2 - 26 = 0 \]
\[ (x - 3)^2 + (y - 6)^2 + (z - 1)^2 = 26 \]

### Step 3: Identify the center of the sphere
From the equation \( (x - 3)^2 + (y - 6)^2 + (z - 1)^2 = 26 \), we can identify the center of the sphere as:
\[ C(3, 6, 1) \]

### Step 4: Use the midpoint formula to find the other end of the diameter
Let \( B(x_1, y_1, z_1) \) be the other end of the diameter. The midpoint \( M \) of the diameter \( AB \) is given by:
\[ M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}, \frac{z_A + z_B}{2} \right) \]
Given \( A(2, 3, 5) \) and \( C(3, 6, 1) \):
\[ M(3, 6, 1) \]

Setting up the equations:
1. For \( x \):
\[ \frac{2 + x_1}{2} = 3 \implies 2 + x_1 = 6 \implies x_1 = 4 \]

2. For \( y \):
\[ \frac{3 + y_1}{2} = 6 \implies 3 + y_1 = 12 \implies y_1 = 9 \]

3. For \( z \):
\[ \frac{5 + z_1}{2} = 1 \implies 5 + z_1 = 2 \implies z_1 = -3 \]

### Step 5: Write down the coordinates of the other end of the diameter
Thus, the coordinates of the other end of the diameter \( B \) are:
\[ B(4, 9, -3) \]

### Final Answer
The coordinates of the other end of the diameter are \( (4, 9, -3) \).

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