Consider a sphere passing through the origin and the points (2,1,-1),(1,5,-4),(-2,4,-6).
Consider the following statements :
1. The sphere passes through the point (0,4,0).
2. The point (1,1,1) is at a distance of 5 unit from the centre of the sphere.
Which of the above statement is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to determine the center of the sphere that passes through the origin and the given points, and then check the validity of the two statements provided.

### 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 sphere's equation
We will substitute the coordinates of the given points into the sphere's equation to form a system of equations.

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

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

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

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

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

From Equation 1, we can express \(w\) in terms of \(u\) and \(v\):
\[ w = 2u + v + 3 \]

Substituting \(w\) into Equations 2 and 3 will allow us to solve for \(u\) and \(v\).

### Step 4: Substitute and solve
Substituting \(w\) into Equation 2:
\[ 2u + 10v - 8(2u + v + 3) = -42 \]
This simplifies to:
\[ 2u + 10v - 16u - 8v - 24 = -42 \implies -14u + 2v = -18 \implies 7u - v = 9 \quad \text{(Equation 4)} \]

Now substituting \(w\) into Equation 3:
\[ -4u + 8v - 12(2u + v + 3) = -56 \]
This simplifies to:
\[ -4u + 8v - 24u - 12v - 36 = -56 \implies -28u - 4v = -20 \implies 7u + v = 5 \quad \text{(Equation 5)} \]

### Step 5: Solve Equations 4 and 5
We have:
1. \(7u - v = 9\) (Equation 4)
2. \(7u + v = 5\) (Equation 5)

Adding these two equations:
\[ (7u - v) + (7u + v) = 9 + 5 \implies 14u = 14 \implies u = 1 \]

Substituting \(u = 1\) back into Equation 4:
\[ 7(1) - v = 9 \implies 7 - v = 9 \implies v = -2 \]

Now substituting \(u = 1\) and \(v = -2\) back into the expression for \(w\):
\[ w = 2(1) + (-2) + 3 = 3 \]

Thus, the center of the sphere is \((-1, -2, 3)\).

### Step 6: Check the statements
1. **Statement 1**: Check if the sphere passes through the point \((0, 4, 0)\).
Substitute into the sphere's equation:
\[ 0^2 + 4^2 + 0^2 + 2(-1)(0) + 2(-2)(4) + 2(3)(0) = 0 \implies 16 - 16 = 0 \]
This is true, so Statement 1 is correct.

2. **Statement 2**: Check the distance from the center \((-1, -2, 3)\) to the point \((1, 1, 1)\).
The distance \(d\) is given by:
\[ d = \sqrt{(1 - (-1))^2 + (1 - (-2))^2 + (1 - 3)^2} = \sqrt{(2)^2 + (3)^2 + (-2)^2} = \sqrt{4 + 9 + 4} = \sqrt{17} \]
Since \(\sqrt{17} \neq 5\), Statement 2 is incorrect.

### Conclusion
The correct answer is:
- Statement 1 is true.
- Statement 2 is false.