The vertices of `DeltaABC` are (2, 0, 0), B(0, 1, 0), C(0, 0, 2). Its orthocentre is H and circumcentre is S. P is a point equidistant from A, B, C and the origin O.
Q. The y-coordinate of S is :

Options

A. `5//6`
B. `1//3`
C. `1//6`
D. `1//2`

Answer

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 will follow these steps:

### Step 1: Identify the coordinates of the vertices
The vertices of the triangle are given as:
- \( A(2, 0, 0) \)
- \( B(0, 1, 0) \)
- \( C(0, 0, 2) \)

### Step 2: Find the equation of the circumcircle
We will consider the line segment \( AB \) as the diameter of the circumcircle. The equation of the circle can be derived using the standard form:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points \( A \) and \( B \).

Substituting \( A(2, 0) \) and \( B(0, 1) \):
\[ (x - 2)(x - 0) + (y - 0)(y - 1) = 0 \]
This simplifies to:
\[ x^2 - 2x + y^2 - y = 0 \]
Let's label this as Equation (1).

### Step 3: Find the equation of line \( AB \)
The slope of line \( AB \) can be calculated as:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{0 - 2} = -\frac{1}{2} \]
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -\frac{1}{2} \) and point \( A(2, 0) \):
\[ y - 0 = -\frac{1}{2}(x - 2) \]
This simplifies to:
\[ x + 2y - 2 = 0 \]
Let's label this as Equation (2).

### Step 4: Find the intersection of the circle and line
To find the circumcenter, we need to solve the system of equations formed by Equation (1) and Equation (2). We substitute Equation (2) into Equation (1):
\[ x^2 - 2x + y^2 - y = 0 \]
Substituting \( y = -\frac{1}{2}x + 1 \) from Equation (2):
\[ x^2 - 2x + \left(-\frac{1}{2}x + 1\right)^2 - \left(-\frac{1}{2}x + 1\right) = 0 \]
Expanding and simplifying gives:
\[ x^2 - 2x + \left(\frac{1}{4}x^2 - x + 1\right) + \frac{1}{2}x - 1 = 0 \]
Combining like terms:
\[ \frac{5}{4}x^2 - \frac{5}{2}x = 0 \]
Factoring out \( x \):
\[ x\left(\frac{5}{4}x - \frac{5}{2}\right) = 0 \]
Thus, \( x = 0 \) or \( x = 2 \).

### Step 5: Find corresponding y-coordinates
For \( x = 0 \):
\[ y = -\frac{1}{2}(0) + 1 = 1 \]
For \( x = 2 \):
\[ y = -\frac{1}{2}(2) + 1 = 0 \]

### Step 6: Find the circumcenter
The circumcenter \( S \) is the average of the y-coordinates:
\[ y_S = \frac{1 + 0}{2} = \frac{1}{2} \]

### Final Answer
The y-coordinate of the circumcenter \( S \) is \( \frac{1}{2} \).

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