Chord joining two distinct point `P(a, 4b) and Q(c, -(16)/(b))` (both are variable points) on the parabola `y^(2)=16x` always passes through a fixed point `(alpha, beta)`. Then, which of the following statements is correct?

Options

A. `alpha+beta=2`
B. `alpha-beta=4`
C. `|alpha|+|beta|=8`
D. `|alpha|=|beta|`

Answer

To solve the problem, we need to analyze the given points \( P(a, 4b) \) and \( Q(c, -\frac{16}{b}) \) on the parabola defined by the equation \( y^2 = 16x \). We will show that the chord joining these two points passes through a fixed point \( (\alpha, \beta) \).

### Step 1: Identify the points on the parabola
The standard form of the parabola \( y^2 = 16x \) can be rewritten as \( y^2 = 4px \) where \( p = 4 \). The coordinates of a point on this parabola can be expressed in terms of a parameter \( t \) as:
\[ (x, y) = (4t^2, 8t) \]

### Step 2: Express points \( P \) and \( Q \) in terms of parameters
For point \( P(a, 4b) \):
- From the parabola, we can set \( 4b = 8t_1 \) which gives \( t_1 = \frac{b}{2} \).
- Therefore, \( a = 4t_1^2 = 4\left(\frac{b}{2}\right)^2 = \frac{b^2}{1} = b^2 \).

For point \( Q(c, -\frac{16}{b}) \):
- From the parabola, we set \( -\frac{16}{b} = 8t_2 \) which gives \( t_2 = -\frac{2}{b} \).
- Therefore, \( c = 4t_2^2 = 4\left(-\frac{2}{b}\right)^2 = \frac{16}{b^2} \).

### Step 3: Find the product of the parameters
The product of the parameters \( t_1 \) and \( t_2 \) is:
\[ t_1 \cdot t_2 = \left(\frac{b}{2}\right) \left(-\frac{2}{b}\right) = -1 \]
This indicates that the chord \( PQ \) is a focal chord of the parabola.

### Step 4: Determine the fixed point through which the chord passes
For a focal chord in a parabola, it always passes through the focus. The focus of the parabola \( y^2 = 16x \) is at the point \( (4, 0) \).

Thus, the fixed point \( (\alpha, \beta) \) is:
\[ (\alpha, \beta) = (4, 0) \]

### Step 5: Analyze the options
Now we can analyze the given options:
1. \( \alpha + \beta = 4 + 0 = 4 \) (Incorrect)
2. \( \alpha - \beta = 4 - 0 = 4 \) (Correct)
3. \( |\alpha| + |\beta| = |4| + |0| = 4 \) (Incorrect)
4. \( |\alpha| = |\beta| \) (Incorrect)

### Conclusion
The correct statement is:
\[ \alpha - \beta = 4 \]