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

Options

A. `(-1,2,-3)`
B. `(1,-2,3)`
C. `(1,1,-3)`
D. `(-1,-2,-3)`

Answer

To find the center 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 passing through the origin can be written as:
\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz = 0 \]
Since the sphere passes through the origin, the constant term is zero.

### Step 2: Substitute the first point (2, 1, -1)
Substituting the coordinates of the first point (2, 1, -1) into the 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 coordinates of the second 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)
Next, substitute the coordinates of the third 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: Solve for \(u\), \(v\), and \(w\)
From Equation 1:
\[ 4u + 2v - 2w = -6 \quad \text{(1)} \]
From Equation 2:
\[ 2u + 10v - 8w = -42 \quad \text{(2)} \]
From Equation 3:
\[ -4u + 8v - 12w = -56 \quad \text{(3)} \]

By solving these equations (using substitution or elimination methods), we find:
- \(u = 1\)
- \(v = -2\)
- \(w = 3\)

### Step 7: Find the center of the sphere
The center of the sphere is given by the coordinates \((-u, -v, -w)\):
\[ \text{Center} = (-1, 2, -3) \]

### Final Answer
The center of the sphere is \((-1, 2, -3)\).

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