If e is the eccentricity of `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` & `theta` be the angle between the asymptotes, then `sec theta//2` equals

Options

A. `e^(2)`
B. `1//e`
C. `2e`
D. e

Answer

To solve the problem, we need to find the value of \( \sec \frac{\theta}{2} \) where \( \theta \) is the angle between the asymptotes of the hyperbola given by the equation:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

### Step 1: Identify the eccentricity of the hyperbola
The eccentricity \( e \) of a hyperbola given by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is given by the formula:

\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]

### Step 2: Determine the angle between the asymptotes
The asymptotes of the hyperbola are given by the equations:

\[ y = \pm \frac{b}{a} x \]

The angle \( \theta \) between the asymptotes can be calculated using the formula:

\[ \theta = 2 \tan^{-1} \left( \frac{b}{a} \right) \]

### Step 3: Calculate \( \tan \frac{\theta}{2} \)
From the angle \( \theta \), we can find \( \tan \frac{\theta}{2} \):

\[ \tan \frac{\theta}{2} = \frac{b}{a} \]

### Step 4: Relate \( \tan \frac{\theta}{2} \) to \( e \)
Now, substituting \( \tan \frac{\theta}{2} \) into the eccentricity formula:

\[ e = \sqrt{1 + \tan^2 \frac{\theta}{2}} \]

### Step 5: Substitute \( \tan \frac{\theta}{2} \) into the eccentricity formula
Using the identity \( 1 + \tan^2 x = \sec^2 x \):

\[ e = \sqrt{1 + \left(\tan \frac{\theta}{2}\right)^2} = \sqrt{1 + \left(\frac{b}{a}\right)^2} \]

### Step 6: Find \( \sec \frac{\theta}{2} \)
Since \( \sec^2 \frac{\theta}{2} = 1 + \tan^2 \frac{\theta}{2} \):

\[ \sec^2 \frac{\theta}{2} = 1 + \left(\frac{b}{a}\right)^2 \]

Taking the square root gives:

\[ \sec \frac{\theta}{2} = \sqrt{1 + \left(\frac{b}{a}\right)^2} \]

### Step 7: Relate \( \sec \frac{\theta}{2} \) to \( e \)
Since we have already established that:

\[ e = \sqrt{1 + \left(\frac{b}{a}\right)^2} \]

Thus, we can conclude:

\[ \sec \frac{\theta}{2} = e \]

### Final Answer
The value of \( \sec \frac{\theta}{2} \) is equal to the eccentricity \( e \).

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