The parabola circumscribing `/_\ABC{(0,0), (4,4), (4,-4)}` and passing through the point `(4, 4)` has the latus rectum

Options

A. 4
B. 1
C. 16
D. 8

Answer

To find the length of the latus rectum of the parabola that circumscribes triangle ABC with vertices at (0,0), (4,4), and (4,-4), and passes through the point (4, 4), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the General Equation of the Parabola**:
The general equation of a parabola that opens to the right is given by:
\[ y^2 = 4ax \]
Here, \(a\) is a parameter that determines the distance from the vertex to the focus.

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

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

4. **Calculate the Length of the Latus Rectum**:
The length of the latus rectum \(L\) of a parabola is given by the formula:
\[ L = 4a \]
Substituting the value of \(a\) we found:
\[ L = 4 \cdot 1 = 4 \]

5. **Conclusion**:
Therefore, the length of the latus rectum of the parabola is:
\[ \boxed{4} \]