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

Options

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

Explanation

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

### Step 2: Substitute the points into the equation
We will substitute each point into the equation to form a system of equations.

#### For the point (2, 1, -1):
\[ 2^2 + 1^2 + (-1)^2 + 2u(2) + 2v(1) + 2w(-1) = 0 \]
This simplifies to:
\[ 4 + 1 + 1 + 4u + 2v - 2w = 0 \implies 4u + 2v - 2w = -6 \quad \text{(Equation 1)} \]

#### For the point (1, 5, -4):
\[ 1^2 + 5^2 + (-4)^2 + 2u(1) + 2v(5) + 2w(-4) = 0 \]
This simplifies to:
\[ 1 + 25 + 16 + 2u + 10v - 8w = 0 \implies 2u + 10v - 8w = -42 \quad \text{(Equation 2)} \]

#### For the point (-2, 4, -6):
\[ (-2)^2 + 4^2 + (-6)^2 + 2u(-2) + 2v(4) + 2w(-6) = 0 \]
This simplifies to:
\[ 4 + 16 + 36 - 4u + 8v - 12w = 0 \implies -4u + 8v - 12w = -56 \quad \text{(Equation 3)} \]

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

#### Combine Equations 1 and 3
Adding Equation 1 and Equation 3:
\[ (4u - 4u) + (2v + 8v) + (-2w - 12w) = -6 - 56 \]
This simplifies to:
\[ 10v - 14w = -62 \quad \text{(Equation 4)} \]

#### Combine Equations 2 and 3
Now, we multiply Equation 2 by 2 and add it to Equation 3:
\[ (2u + 10v - 8w) \times 2 + (-4u + 8v - 12w) = -42 \times 2 - 56 \]
This gives:
\[ (4u - 4u) + (20v + 8v) + (-16w - 12w) = -84 - 56 \]
This simplifies to:
\[ 28v - 28w = -140 \quad \text{(Equation 5)} \]

### Step 4: Solve for \(v\) and \(w\)
Now we have two new equations:
1. \(10v - 14w = -62\) (Equation 4)
2. \(28v - 28w = -140\) (Equation 5)

From Equation 5, we can simplify:
\[ v - w = -5 \quad \text{(Equation 6)} \]

Substituting \(w = v + 5\) into Equation 4:
\[ 10v - 14(v + 5) = -62 \]
This simplifies to:
\[ 10v - 14v - 70 = -62 \implies -4v = 8 \implies v = -2 \]

### Step 5: Find \(w\) and \(u\)
Using \(v = -2\) in Equation 6:
\[ w = -2 + 5 = 3 \]

Now substitute \(v\) and \(w\) back into Equation 1 to find \(u\):
\[ 4u + 2(-2) - 2(3) = -6 \]
This simplifies to:
\[ 4u - 4 - 6 = -6 \implies 4u = 4 \implies u = 1 \]

### Step 6: Find the center of the sphere
The center of the sphere is given by:
\[ (-u, -v, -w) = (-1, 2, -3) \]

Thus, the center of the sphere is:
\[ \boxed{(-1, 2, -3)} \]