What is the radius of the sphere `x^2+y^2+z^2-6x+8y-10z+1=0`?

Options

A. 5
B. 2
C. 7
D. 3

Explanation

To find the radius of the sphere given by the equation \( x^2 + y^2 + z^2 - 6x + 8y - 10z + 1 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation in standard form
The standard form of the equation of a sphere is given by:
\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz - d = 0 \]
We need to rearrange the given equation to match this form.

### Step 2: Identify coefficients
From the given equation:
\[ x^2 + y^2 + z^2 - 6x + 8y - 10z + 1 = 0 \]
we can identify:
- \(2u = -6\) (so \(u = -3\))
- \(2v = 8\) (so \(v = 4\))
- \(2w = -10\) (so \(w = -5\))
- \(d = -1\)

### Step 3: Apply the radius formula
The formula for the radius \( r \) of the sphere is:
\[ r = \sqrt{u^2 + v^2 + w^2 - d} \]
Now we substitute the values we found:
- \(u = -3\)
- \(v = 4\)
- \(w = -5\)
- \(d = -1\)

### Step 4: Calculate \( u^2 + v^2 + w^2 - d \)
Now we calculate:
\[ u^2 = (-3)^2 = 9 \]
\[ v^2 = 4^2 = 16 \]
\[ w^2 = (-5)^2 = 25 \]
Now, substituting these into the formula:
\[ r = \sqrt{9 + 16 + 25 - (-1)} \]
\[ r = \sqrt{9 + 16 + 25 + 1} \]
\[ r = \sqrt{51} \]

### Step 5: Final calculation
Now we compute:
\[ r = \sqrt{51} \]

### Conclusion
Thus, the radius of the sphere is:
\[ \boxed{7} \]