Consider a sphere passing through the origin and the points `(2,1,-1),(1,5,-4),(-2,4,-6)`
What is the radius of the sphere

Options

A. `sqrt(12)`
B. `sqrt(14)`
C. 12
D. 14

Answer

To find the radius of the sphere that passes through the origin and the points (2, 1, -1), (1, 5, -4), and (-2, 4, -6), we can follow these steps:

### Step 1: Write the equation of the sphere
The general equation of a sphere that passes through the origin is given by:
\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz = 0 \]
where \( (u, v, w) \) are the coefficients we need to determine.

### Step 2: Substitute the first point (2, 1, -1)
Substituting the point (2, 1, -1) into the sphere's equation:
\[ 2^2 + 1^2 + (-1)^2 + 2u(2) + 2v(1) + 2w(-1) = 0 \]
Calculating this gives:
\[ 4 + 1 + 1 + 4u + 2v - 2w = 0 \]
This simplifies to:
\[ 4u + 2v - 2w + 6 = 0 \quad \text{(Equation 1)} \]

### Step 3: Substitute the second point (1, 5, -4)
Now, substitute the point (1, 5, -4):
\[ 1^2 + 5^2 + (-4)^2 + 2u(1) + 2v(5) + 2w(-4) = 0 \]
Calculating this gives:
\[ 1 + 25 + 16 + 2u + 10v - 8w = 0 \]
This simplifies to:
\[ 2u + 10v - 8w + 42 = 0 \quad \text{(Equation 2)} \]

### Step 4: Substitute the third point (-2, 4, -6)
Now, substitute the point (-2, 4, -6):
\[ (-2)^2 + 4^2 + (-6)^2 + 2u(-2) + 2v(4) + 2w(-6) = 0 \]
Calculating this gives:
\[ 4 + 16 + 36 - 4u + 8v - 12w = 0 \]
This simplifies to:
\[ -4u + 8v - 12w + 56 = 0 \quad \text{(Equation 3)} \]

### Step 5: Solve the system of equations
Now we have a system of three equations:
1. \( 4u + 2v - 2w + 6 = 0 \)
2. \( 2u + 10v - 8w + 42 = 0 \)
3. \( -4u + 8v - 12w + 56 = 0 \)

We can solve these equations simultaneously to find the values of \( u, v, \) and \( w \).

### Step 6: Find the values of u, v, and w
From the first equation, we can express \( w \):
\[ w = 2u + v + 3 \]
Substituting \( w \) into the second and third equations will allow us to solve for \( u \) and \( v \).

After solving, we find:
- \( u = 1 \)
- \( v = -2 \)
- \( w = 3 \)

### Step 7: Calculate the radius of the sphere
The radius \( r \) of the sphere is given by:
\[ r = \sqrt{u^2 + v^2 + w^2} \]
Substituting the values we found:
\[ r = \sqrt{1^2 + (-2)^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Final Answer
The radius of the sphere is \( \sqrt{14} \).

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