If eccentricity of the hyperbola `(x^(2))/(cos^(2) theta)-(y^(2))/(sin^(2) theta)=1` is move than `2` when `theta in (0,(pi)/2)`. Find the possible values of length of latus rectum (a) `(3,oo)` (b) `1,3//2)` (c) `(2,3)` (d) `(-3,-2)`

Options

A. (2,3)
B. `(3, oo)`
C. `(3//2, 2)`
D. `(1, 3//2)`

Answer

To solve the problem, we need to analyze the given hyperbola and its properties step by step.

### Step 1: Identify the equation of the hyperbola
The given equation of the hyperbola is:
\[ \frac{x^2}{\cos^2 \theta} - \frac{y^2}{\sin^2 \theta} = 1 \]
Here, we can identify \( a^2 = \cos^2 \theta \) and \( b^2 = \sin^2 \theta \).

### Step 2: Find the eccentricity of the hyperbola
The eccentricity \( e \) of a hyperbola is given by the formula:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ e = \sqrt{1 + \frac{\sin^2 \theta}{\cos^2 \theta}} = \sqrt{1 + \tan^2 \theta} \]
Using the identity \( 1 + \tan^2 \theta = \sec^2 \theta \):
\[ e = \sec \theta \]

### Step 3: Set up the inequality for eccentricity
According to the problem, the eccentricity is more than 2:
\[ e > 2 \implies \sec \theta > 2 \]
This implies:
\[ \cos \theta < \frac{1}{2} \]

### Step 4: Determine the range of \( \theta \)
The condition \( \cos \theta < \frac{1}{2} \) corresponds to the angles:
\[ \theta \in \left(\frac{\pi}{3}, \frac{\pi}{2}\right) \]

### Step 5: Calculate the length of the latus rectum
The length of the latus rectum \( L \) of a hyperbola is given by:
\[ L = \frac{2b^2}{a} \]
Substituting \( a = \cos \theta \) and \( b = \sin \theta \):
\[ L = \frac{2 \sin^2 \theta}{\cos \theta} \]
This can be rewritten using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \):
\[ L = 2 \frac{\sin^2 \theta}{\cos \theta} = 2 \tan \theta \sin \theta \]

### Step 6: Analyze the expression for \( L \)
Using the interval \( \theta \in \left(\frac{\pi}{3}, \frac{\pi}{2}\right) \):
- At \( \theta = \frac{\pi}{3} \):
\[ L = 2 \tan\left(\frac{\pi}{3}\right) \sin\left(\frac{\pi}{3}\right) = 2 \cdot \sqrt{3} \cdot \frac{\sqrt{3}}{2} = 3 \]
- As \( \theta \) approaches \( \frac{\pi}{2} \), \( \tan \theta \) approaches infinity, thus \( L \) approaches infinity.

### Conclusion
The possible values of the length of the latus rectum \( L \) are:
\[ L \in (3, \infty) \]
Thus, the correct answer is:
**(a) (3, ∞)**