If `alpha and beta` are two points on the hyperbola `x^(2)/a^(2)-y^(2)/b^(2)=1` and the chord joining these two points passes through the focus (ae, 0) then `e cos ""(alpha-beta)/(2)=`

Options

A. a) `cos ""(alpha+beta)/(2)`
B. b) `cos""(alpha-beta)/(2)`
C. c) `cos""(2alpha+2beta)/(4)`
D. d) `cos""(alpha+beta)/(4)`

Answer

To solve the problem, we need to find the value of \( e \cos \frac{\alpha - \beta}{2} \) given that the chord joining two points on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) passes through the focus \( (ae, 0) \).

### Step-by-Step Solution:

1. **Identify Points on the Hyperbola**:
Let the two points on the hyperbola be represented as:
- Point \( P \): \( (a \sec \alpha, b \tan \alpha) \)
- Point \( Q \): \( (a \sec \beta, b \tan \beta) \)

2. **Equation of the Chord**:
The equation of the chord joining points \( P \) and \( Q \) can be derived using the two-point form of the line equation:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]
Substituting the coordinates of points \( P \) and \( Q \):
\[ y - b \tan \beta = \frac{b \tan \alpha - b \tan \beta}{a \sec \alpha - a \sec \beta} (x - a \sec \beta) \]

3. **Simplifying the Equation**:
The slope of the chord can be simplified as:
\[ \text{slope} = \frac{b (\tan \alpha - \tan \beta)}{a (\sec \alpha - \sec \beta)} \]
After simplification, the equation of the chord becomes:
\[ \frac{x}{a} \cos \frac{\alpha - \beta}{2} - \frac{y}{b} \sin \frac{\alpha + \beta}{2} = \cos \frac{\alpha + \beta}{2} \]

4. **Substituting the Focus**:
Since the chord passes through the focus \( (ae, 0) \), we substitute \( x = ae \) and \( y = 0 \) into the chord equation:
\[ \frac{ae}{a} \cos \frac{\alpha - \beta}{2} - \frac{0}{b} \sin \frac{\alpha + \beta}{2} = \cos \frac{\alpha + \beta}{2} \]
This simplifies to:
\[ e \cos \frac{\alpha - \beta}{2} = \cos \frac{\alpha + \beta}{2} \]

5. **Final Result**:
Thus, we have:
\[ e \cos \frac{\alpha - \beta}{2} = \cos \frac{\alpha + \beta}{2} \]

### Conclusion:
The value of \( e \cos \frac{\alpha - \beta}{2} \) is \( \cos \frac{\alpha + \beta}{2} \).