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

Answer

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: Write the general equation of a sphere
The general equation of a sphere is given by:
\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 \]
where \( (u, v, w) \) are the coordinates of the center and \( d \) is a constant.

### Step 2: Compare the given equation with the general equation
We can rewrite the given equation:
\[ x^2 + y^2 + z^2 - 6x + 8y - 10z + 1 = 0 \]
Now, we can identify the coefficients:
- \( 2u = -6 \) → \( u = -3 \)
- \( 2v = 8 \) → \( v = 4 \)
- \( 2w = -10 \) → \( w = -5 \)
- \( d = 1 \)

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

### Step 4: Substitute the values into the formula
Now, substitute the values of \( u \), \( v \), \( w \), and \( d \) into the formula:
\[ r = \sqrt{(-3)^2 + (4)^2 + (-5)^2 - 1} \]

### Step 5: Calculate the squares and simplify
Calculating the squares:
- \( (-3)^2 = 9 \)
- \( (4)^2 = 16 \)
- \( (-5)^2 = 25 \)

Now substitute these values:
\[ r = \sqrt{9 + 16 + 25 - 1} \]
\[ r = \sqrt{49} \]

### Step 6: Find the radius
Taking the square root:
\[ r = 7 \]

Thus, the radius of the sphere is \( 7 \) units.

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