To solve the problem, we need to analyze the points \( P(a, 4b) \) and \( Q(c, -\frac{16}{b}) \) on the parabola given by the equation \( y^2 = 16x \). We will determine the conditions under which the chord joining these points always passes through a fixed point \( (\alpha, \beta) \).
### Step-by-Step Solution:
1. **Identify the points on the parabola**:
The parabola \( y^2 = 16x \) can be parameterized as:
\[
(x, y) = (4t^2, 8t)
\]
For point \( P(a, 4b) \):
- We set \( 4b = 8t_1 \) which gives \( t_1 = \frac{b}{2} \).
- Thus, \( P \) can be expressed as \( P(4t_1^2, 8t_1) = \left(4\left(\frac{b}{2}\right)^2, 4b\right) = \left(b^2, 4b\right) \).
2. **For point \( Q(c, -\frac{16}{b}) \)**:
- We set \( -\frac{16}{b} = 8t_2 \) which gives \( t_2 = -\frac{2}{b} \).
- Thus, \( Q \) can be expressed as \( Q(4t_2^2, 8t_2) = \left(4\left(-\frac{2}{b}\right)^2, -\frac{16}{b}\right) = \left(\frac{16}{b^2}, -\frac{16}{b}\right) \).
3. **Finding the equation of the chord PQ**:
The slope \( m \) of the line joining points \( P \) and \( Q \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-\frac{16}{b} - 4b}{\frac{16}{b^2} - b^2}
\]
4. **Equation of the line**:
The equation of the line in point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( P(b^2, 4b) \) into the equation gives:
\[
y - 4b = m(x - b^2)
\]
5. **Condition for the line to pass through a fixed point \( (\alpha, \beta) \)**:
For the line to always pass through the fixed point \( (\alpha, \beta) \), we must have:
\[
\beta - 4b = m(\alpha - b^2)
\]
This must hold for all values of \( b \).
6. **Finding the fixed point**:
If we analyze the equation, we can see that for the line to pass through a fixed point regardless of \( b \), the coefficients of \( b \) must cancel out. This leads us to conclude that the fixed point must be the focus of the parabola, which is at \( (4, 0) \).
7. **Conclusion**:
Therefore, the fixed point \( (\alpha, \beta) \) is \( (4, 0) \).
### Final Answer:
The correct statement is that the fixed point through which the chord passes is \( (4, 0) \).
