To find the y-coordinate of the circumcenter \( S \) of triangle \( \Delta ABC \) with vertices \( A(2, 0, 0) \), \( B(0, 1, 0) \), and \( C(0, 0, 2) \), we can follow these steps:
### Step 1: Find the Midpoint of Segment \( BC \)
The coordinates of points \( B \) and \( C \) are:
- \( B(0, 1, 0) \)
- \( C(0, 0, 2) \)
The midpoint \( D \) of segment \( BC \) can be calculated using the midpoint formula:
\[
D = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, \frac{z_B + z_C}{2} \right)
\]
Substituting the coordinates:
\[
D = \left( \frac{0 + 0}{2}, \frac{1 + 0}{2}, \frac{0 + 2}{2} \right) = (0, \frac{1}{2}, 1)
\]
**Hint:** The midpoint formula is useful for finding the center of a line segment.
### Step 2: Find the Direction Ratios of \( SD \)
Let the coordinates of circumcenter \( S \) be \( (a, b, c) \). The direction ratios of line segment \( SD \) can be expressed as:
\[
\text{Direction Ratios of } SD = (a - 0, b - \frac{1}{2}, c - 1) = (a, b - \frac{1}{2}, c - 1)
\]
**Hint:** Direction ratios represent the change in coordinates from one point to another.
### Step 3: Find the Direction Ratios of Segment \( BC \)
The direction ratios of segment \( BC \) can be calculated as:
\[
\text{Direction Ratios of } BC = (0 - 0, 0 - 1, 2 - 0) = (0, -1, 2)
\]
**Hint:** The direction ratios help in determining the orientation of a line segment.
### Step 4: Set Up the Perpendicular Condition
Since \( SD \) is perpendicular to \( BC \), their dot product must equal zero:
\[
a \cdot 0 + (b - \frac{1}{2})(-1) + (c - 1)(2) = 0
\]
This simplifies to:
\[
-(b - \frac{1}{2}) + 2(c - 1) = 0
\]
Rearranging gives:
\[
b - \frac{1}{2} = 2c - 2
\]
Thus,
\[
b = 2c - \frac{3}{2}
\]
**Hint:** The dot product of two perpendicular vectors is zero.
### Step 5: Find the Midpoint of Segment \( AC \)
Next, we find the midpoint \( E \) of segment \( AC \):
\[
E = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}, \frac{z_A + z_C}{2} \right)
\]
Substituting the coordinates of \( A \) and \( C \):
\[
E = \left( \frac{2 + 0}{2}, \frac{0 + 0}{2}, \frac{0 + 2}{2} \right) = (1, 0, 1)
\]
**Hint:** Finding midpoints helps in determining key points in geometric figures.
### Step 6: Find the Direction Ratios of Segment \( SE \)
The direction ratios of segment \( SE \) are:
\[
\text{Direction Ratios of } SE = (a - 1, b - 0, c - 1) = (a - 1, b, c - 1)
\]
**Hint:** Direction ratios can also be used to analyze relationships between different segments.
### Step 7: Find the Direction Ratios of Segment \( AC \)
The direction ratios of segment \( AC \) are:
\[
\text{Direction Ratios of } AC = (2 - 0, 0 - 0, 0 - 2) = (2, 0, -2)
\]
**Hint:** Understanding the direction ratios of segments helps in establishing relationships.
### Step 8: Set Up the Perpendicular Condition for \( SE \)
Since \( SE \) is perpendicular to \( AC \), we set up the dot product condition:
\[
(a - 1) \cdot 2 + b \cdot 0 + (c - 1)(-2) = 0
\]
This simplifies to:
\[
2(a - 1) - 2(c - 1) = 0
\]
Rearranging gives:
\[
2a - 2 - 2c + 2 = 0 \implies 2a = 2c \implies a = c
\]
**Hint:** The conditions for perpendicularity can lead to important relationships between coordinates.
### Step 9: Substitute \( a = c \) into the Equation for \( b \)
Now substitute \( a \) for \( c \) in the equation \( b = 2c - \frac{3}{2} \):
\[
b = 2a - \frac{3}{2}
\]
**Hint:** Substituting known values simplifies the equations.
### Step 10: Find the Equation of the Plane
The equation of the plane containing points \( A, B, C \) is given by:
\[
\frac{x}{2} + \frac{y}{1} + \frac{z}{2} = 1
\]
Substituting \( (a, b, c) \) into the plane equation:
\[
\frac{a}{2} + \frac{b}{1} + \frac{c}{2} = 1
\]
Substituting \( b = 2a - \frac{3}{2} \) and \( c = a \):
\[
\frac{a}{2} + (2a - \frac{3}{2}) + \frac{a}{2} = 1
\]
This simplifies to:
\[
2a - \frac{3}{2} = 1 \implies 2a = \frac{5}{2} \implies a = \frac{5}{4}
\]
**Hint:** Solving equations systematically leads to the final values.
### Step 11: Find \( b \)
Now substituting \( a \) back to find \( b \):
\[
b = 2 \cdot \frac{5}{4} - \frac{3}{2} = \frac{10}{4} - \frac{6}{4} = \frac{4}{4} = 1
\]
**Hint:** Always check your calculations to ensure accuracy.
### Conclusion
The y-coordinate of the circumcenter \( S \) is:
\[
\boxed{1}
\]
