Tangents are drawn at the ends of any focal chord of the parabola `y^(2)=16x`. Then which of the following statements about the point of intersection of tangents is true.

Options

A. its abscissa is independent of the extremities of the focal chord
B. its ordinate is independent of the extremities of the focal chord
C. it is at a distance of 8 units from the vertex of the parabola
D. It is at a distance of 16 units from the focus of the parabola

Answer

To solve the problem, we need to analyze the tangents drawn at the ends of any focal chord of the parabola given by the equation \( y^2 = 16x \).

### Step-by-Step Solution:

1. **Identify the Parabola and its Properties**:
The given parabola is \( y^2 = 16x \). This is a standard form of a parabola that opens to the right. The vertex of this parabola is at the origin (0,0), and the focus is at the point (4,0).

**Hint**: Remember that the standard form of a parabola \( y^2 = 4ax \) helps identify the focus and directrix.

2. **Determine the Focal Chord**:
A focal chord is a line segment that passes through the focus of the parabola. Let’s denote the endpoints of the focal chord as \( P(t_1) \) and \( P(t_2) \) where \( t_1 \) and \( t_2 \) are parameters such that \( t_1 t_2 = -1 \). The coordinates of these points can be expressed as:
\[ P(t_1) = (4t_1^2, 8t_1) \quad \text{and} \quad P(t_2) = (4t_2^2, 8t_2) \]

**Hint**: Use the parameterization of the parabola to find the coordinates of points on the parabola.

3. **Find the Tangents at the Endpoints**:
The equation of the tangent to the parabola \( y^2 = 16x \) at the point \( (4t^2, 8t) \) is given by:
\[ yy_1 = 8(x + x_1) \]
where \( (x_1, y_1) \) is the point of tangency. Thus, the tangents at points \( P(t_1) \) and \( P(t_2) \) can be written as:
\[ yy_1 = 8(x + 4t_1^2) \quad \text{and} \quad yy_2 = 8(x + 4t_2^2) \]

**Hint**: Recall the formula for the tangent to a parabola at a given point.

4. **Find the Point of Intersection of the Tangents**:
To find the point of intersection of the two tangents, we need to solve the equations simultaneously. Setting the two equations equal to each other will give us the coordinates of the intersection point \( R \).

After substituting and simplifying, we find that the x-coordinate of the point of intersection \( R \) is:
\[ x = -4 \]
The y-coordinate can be derived from the tangents, but we focus on the x-coordinate for now.

**Hint**: When solving simultaneous equations, isolate one variable to simplify the process.

5. **Analyze the Results**:
The x-coordinate of the point of intersection is independent of the specific values of \( t_1 \) and \( t_2 \) (as long as they satisfy the condition \( t_1 t_2 = -1 \)). This means that regardless of the focal chord chosen, the x-coordinate of the intersection point remains constant.

**Hint**: Check if the derived coordinates depend on the parameters chosen.

6. **Conclusion**:
The correct statement about the point of intersection of the tangents drawn at the ends of any focal chord of the parabola \( y^2 = 16x \) is that the x-coordinate of the point of intersection is independent of the extremities of the focal chord.

### Final Answer:
The point of intersection of the tangents is independent of the extremities of the focal chord.