The parabola circumscribing `/_\ABC` and passing through the point `(4, 4)` has the focus

Options

A. `(0,1)`
B. `(4,0)`
C. `(1,0)`
D. `(0,4)`

Answer

To solve the problem, we need to find the coordinates of the focus of a parabola that circumscribes triangle ABC and passes through the point (4, 4). We will assume the parabola is horizontal.

### Step-by-Step Solution:

1. **Identify the Form of the Parabola**:
The standard equation for a horizontal parabola is given by:
\[ y^2 = 4ax \]
where \(a\) is the distance from the vertex to the focus.

2. **Substitute the Given Point**:
Since the parabola passes through the point (4, 4), we can substitute \(x = 4\) and \(y = 4\) into the equation:
\[ 4^2 = 4a(4) \]
This simplifies to:
\[ 16 = 16a \]

3. **Solve for \(a\)**:
Dividing both sides by 16 gives:
\[ a = 1 \]

4. **Determine the Coordinates of the Focus**:
For a horizontal parabola, the coordinates of the focus are given by \((a, 0)\). Since we found \(a = 1\), the coordinates of the focus are:
\[ (1, 0) \]

5. **Final Answer**:
The coordinates of the focus of the parabola are:
\[ \boxed{(1, 0)} \]