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

Explanation

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

### Step 1: Rewrite the equation in standard form
The general equation of a sphere is given by:
\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 \]
Comparing this with the given equation:
\[ x^2 + y^2 + z^2 - 4x + 6y - 8z - 7 = 0 \]
we can identify:
- \(2u = -4\) → \(u = -2\)
- \(2v = 6\) → \(v = 3\)
- \(2w = -8\) → \(w = -4\)
- \(d = -7\)

### Step 2: Use the formula for the radius of the sphere
The radius \(r\) of the sphere can be calculated using the formula:
\[ r = \sqrt{u^2 + v^2 + w^2 - d} \]
Substituting the values we found:
\[ r = \sqrt{(-2)^2 + 3^2 + (-4)^2 - (-7)} \]

### Step 3: Calculate the values inside the square root
Calculating each term:
- \((-2)^2 = 4\)
- \(3^2 = 9\)
- \((-4)^2 = 16\)
- \(-(-7) = 7\)

Now, adding these together:
\[ r = \sqrt{4 + 9 + 16 + 7} = \sqrt{36} \]

### Step 4: Find the radius
Calculating the square root:
\[ r = 6 \]

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

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

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