`y^(2)=16x` is a parabola and `x^(2)+y^(2)=16` is a circle . Then

Options

A. circle passes through the vertex of the parabola
B. circle touches the parabola at the vertex
C. circle passes through the focus of the parabola
D. circle lies inside the parabola .

Answer

To solve the problem, we need to analyze the given equations of the parabola and the circle, and determine the relationship between them.

### Step-by-Step Solution:

1. **Identify the equations**:
- The equation of the parabola is given as \( y^2 = 16x \).
- The equation of the circle is given as \( x^2 + y^2 = 16 \).

2. **Convert the parabola to standard form**:
- The standard form of a parabola is \( y^2 = 4ax \).
- Here, we can see that \( 4a = 16 \), so \( a = 4 \).
- The focus of the parabola, which is at the point \( (a, 0) \), is therefore \( (4, 0) \).

3. **Identify the circle's center and radius**:
- The equation of the circle \( x^2 + y^2 = r^2 \) indicates that the center is at \( (0, 0) \) and the radius \( r \) is \( \sqrt{16} = 4 \).

4. **Find the vertex of the parabola**:
- The vertex of the parabola \( y^2 = 16x \) is at the origin \( (0, 0) \).

5. **Check if the circle passes through the vertex of the parabola**:
- Substitute the vertex \( (0, 0) \) into the circle's equation:
\[ 0^2 + 0^2 = 16 \implies 0 \neq 16 \]
- Thus, the circle does not pass through the vertex of the parabola.

6. **Check if the circle touches the parabola at the vertex**:
- Since the circle does not pass through the vertex, it cannot touch the parabola at that point.

7. **Check if the circle passes through the focus of the parabola**:
- The focus of the parabola is \( (4, 0) \).
- Substitute \( (4, 0) \) into the circle's equation:
\[ 4^2 + 0^2 = 16 \implies 16 = 16 \]
- Therefore, the circle does pass through the focus of the parabola.

8. **Check if the circle is inside the parabola**:
- The parabola opens to the right, and since the radius of the circle is 4 and the focus is at (4, 0), the circle cannot be entirely inside the parabola.

### Conclusion:
- The circle does not pass through the vertex of the parabola.
- The circle does not touch the parabola at the vertex.
- The circle passes through the focus of the parabola.
- The circle is not inside the parabola.

Thus, the correct option is that the circle passes through the focus of the parabola.