What is the diameter of the sphere `x^2+y^2+z^2-4x+6y-8z-7=0`?

Options

A. 4 units
B. 5 units
C. 6 units
D. 12 units

Answer

To find the diameter of the sphere given by the equation \(x^2 + y^2 + z^2 - 4x + 6y - 8z - 7 = 0\), we can follow these steps:

### Step 1: Rewrite the equation in standard form
The standard form of the equation of a sphere is:
\[ (x - u)^2 + (y - v)^2 + (z - w)^2 = r^2 \]
We start by rearranging the given equation:
\[ x^2 - 4x + y^2 + 6y + z^2 - 8z - 7 = 0 \]

### Step 2: Complete the square for each variable
We will complete the square for \(x\), \(y\), and \(z\).

1. For \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

2. For \(y\):
\[ y^2 + 6y = (y + 3)^2 - 9 \]

3. For \(z\):
\[ z^2 - 8z = (z - 4)^2 - 16 \]

### Step 3: Substitute back into the equation
Substituting these completed squares back into the equation gives:
\[ ((x - 2)^2 - 4) + ((y + 3)^2 - 9) + ((z - 4)^2 - 16) - 7 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y + 3)^2 + (z - 4)^2 - 36 = 0 \]
Thus, we have:
\[ (x - 2)^2 + (y + 3)^2 + (z - 4)^2 = 36 \]

### Step 4: Identify the center and radius
From the standard form, we can see that:
- The center of the sphere is \((u, v, w) = (2, -3, 4)\).
- The radius \(r\) is given by \(r^2 = 36\), so \(r = \sqrt{36} = 6\).

### Step 5: Calculate the diameter
The diameter \(D\) of the sphere is given by:
\[ D = 2r = 2 \times 6 = 12 \]

### Final Answer
Thus, the diameter of the sphere is \(12\).

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