To solve the problem, we need to find the equation of the sphere that passes through the origin and the points (2, 1, -1), (1, 5, -4), and (-2, 4, -6). We can then verify the two statements provided.
### 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 coordinates of the center of the sphere.
### Step 2: Substitute the points into the sphere's equation
We will substitute each of the three points into the equation to form a system of equations.
**For the point (2, 1, -1):**
\[
2^2 + 1^2 + (-1)^2 + 2u(2) + 2v(1) + 2w(-1) = 0
\]
\[
4 + 1 + 1 + 4u + 2v - 2w = 0
\]
\[
6 + 4u + 2v - 2w = 0 \quad \text{(Equation 1)}
\]
**For the point (1, 5, -4):**
\[
1^2 + 5^2 + (-4)^2 + 2u(1) + 2v(5) + 2w(-4) = 0
\]
\[
1 + 25 + 16 + 2u + 10v - 8w = 0
\]
\[
42 + 2u + 10v - 8w = 0 \quad \text{(Equation 2)}
\]
**For the point (-2, 4, -6):**
\[
(-2)^2 + 4^2 + (-6)^2 + 2u(-2) + 2v(4) + 2w(-6) = 0
\]
\[
4 + 16 + 36 - 4u + 8v - 12w = 0
\]
\[
56 - 4u + 8v - 12w = 0 \quad \text{(Equation 3)}
\]
### Step 3: Solve the system of equations
We now have three equations:
1. \( 4u + 2v - 2w = -6 \)
2. \( 2u + 10v - 8w = -42 \)
3. \( -4u + 8v - 12w = -56 \)
We can solve these equations simultaneously to find the values of \( u, v, \) 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:
\[
-4u + 8v - 12w = -56 \quad \text{(3)}
\]
We can express \( w \) in terms of \( u \) and \( v \) from Equation (1):
\[
2w = 4u + 2v + 6 \implies w = 2u + v + 3
\]
Substituting \( w \) into Equations (2) and (3) will allow us to solve for \( u \) and \( v \).
### Step 4: Substitute \( w \) into the equations
Substituting \( w \) into Equation (2):
\[
2u + 10v - 8(2u + v + 3) = -42
\]
\[
2u + 10v - 16u - 8v - 24 = -42
\]
\[
-14u + 2v + 24 = -42
\]
\[
-14u + 2v = -66 \implies 7u - v = 33 \quad \text{(4)}
\]
Substituting \( w \) into Equation (3):
\[
-4u + 8v - 12(2u + v + 3) = -56
\]
\[
-4u + 8v - 24u - 12v - 36 = -56
\]
\[
-28u - 4v - 36 = -56
\]
\[
-28u - 4v = -20 \implies 7u + v = 5 \quad \text{(5)}
\]
### Step 5: Solve Equations (4) and (5)
Now we have:
1. \( 7u - v = 33 \)
2. \( 7u + v = 5 \)
Adding these two equations:
\[
(7u - v) + (7u + v) = 33 + 5
\]
\[
14u = 38 \implies u = \frac{19}{7}
\]
Substituting \( u \) back into Equation (4):
\[
7 \left( \frac{19}{7} \right) - v = 33 \implies 19 - v = 33 \implies v = -14
\]
Substituting \( u \) and \( v \) back into the expression for \( w \):
\[
w = 2 \left( \frac{19}{7} \right) + (-14) + 3 = \frac{38}{7} - 14 + 3 = \frac{38}{7} - \frac{98}{7} + \frac{21}{7} = \frac{-39}{7}
\]
### Step 6: Write the equation of the sphere
Now we have \( u = \frac{19}{7}, v = -14, w = \frac{-39}{7} \). The equation of the sphere becomes:
\[
x^2 + y^2 + z^2 + 2 \left( \frac{19}{7} \right)x - 28y - \frac{78}{7}z = 0
\]
### Step 7: Verify the statements
**Statement I: The sphere passes through the point (0, 4, 0)**
Substituting \( (0, 4, 0) \) into the sphere's equation:
\[
0 + 16 + 0 + 0 - 28(4) + 0 = 0
\]
\[
16 - 112 = -96 \neq 0 \quad \text{(False)}
\]
**Statement II: The point (1, 1, 1) is at a distance of 5 units from the center of the sphere**
The center of the sphere is \( (-u, -v, -w) = \left(-\frac{19}{7}, 14, \frac{39}{7}\right) \).
Calculating the distance from \( (1, 1, 1) \):
\[
d = \sqrt{\left(1 + \frac{19}{7}\right)^2 + \left(1 - 14\right)^2 + \left(1 - \frac{39}{7}\right)^2}
\]
Calculating each term:
1. \( 1 + \frac{19}{7} = \frac{26}{7} \)
2. \( 1 - 14 = -13 \)
3. \( 1 - \frac{39}{7} = \frac{-32}{7} \)
Now calculating the distance:
\[
d = \sqrt{\left(\frac{26}{7}\right)^2 + (-13)^2 + \left(\frac{-32}{7}\right)^2}
\]
\[
= \sqrt{\frac{676}{49} + 169 + \frac{1024}{49}} = \sqrt{\frac{676 + 169 \cdot 49 + 1024}{49}} = \sqrt{\frac{676 + 8281 + 1024}{49}} = \sqrt{\frac{9981}{49}} = \frac{\sqrt{9981}}{7}
\]
Calculating \( \sqrt{9981} \) gives approximately \( 99.905 \), thus:
\[
d \approx \frac{99.905}{7} \approx 14.27 \quad \text{(False)}
\]
### Conclusion
Both statements are incorrect.
**Final Answer:**
Neither statement I nor II is correct.
