Consider the equation
`sec theta +cosec theta=a, theta in (0, 2pi) -{pi//2, pi, 3pi//2}`
If the equation has four distinct real roots, then (a) `|a| gt 2sqrt(2)` (b) `|a| lt 2sqrt(2)` (c) `a ge -2sqrt(2)` (d) none of these

Options

A. `|a| gt 2sqrt(2)`
B. `|a| lt 2sqrt(2)`
C. `a ge -2sqrt(2)`
D. none of these

Explanation

To solve the equation \( \sec \theta + \csc \theta = a \) for \( \theta \in (0, 2\pi) - \{\frac{\pi}{2}, \pi, \frac{3\pi}{2}\} \), we need to analyze the behavior of the left-hand side of the equation and determine the conditions under which it can have four distinct real roots.

### Step 1: Rewrite the equation
We start by rewriting the equation in terms of sine and cosine:
\[ \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta} \]
Thus, the equation becomes:
\[ \frac{1}{\cos \theta} + \frac{1}{\sin \theta} = a \]

### Step 2: Combine the fractions
We can combine the fractions on the left-hand side:
\[ \frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta} = a \]
This leads to:
\[ \sin \theta + \cos \theta = a \sin \theta \cos \theta \]

### Step 3: Analyze the function \( f(\theta) = \sin \theta + \cos \theta \)
The function \( f(\theta) = \sin \theta + \cos \theta \) can be rewritten using the identity:
\[ f(\theta) = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
The maximum value of \( f(\theta) \) is \( \sqrt{2} \) and the minimum value is \( -\sqrt{2} \).

### Step 4: Determine the range of \( a \)
For the equation \( \sin \theta + \cos \theta = a \sin \theta \cos \theta \) to have four distinct real roots, the value of \( a \) must be such that the right-hand side can take on values in the range of \( f(\theta) \).

The product \( \sin \theta \cos \theta \) can be rewritten as:
\[ \sin \theta \cos \theta = \frac{1}{2} \sin(2\theta) \]
The maximum value of \( \sin(2\theta) \) is 1, so the maximum value of \( \sin \theta \cos \theta \) is \( \frac{1}{2} \).

### Step 5: Set conditions for \( a \)
For \( a \sin \theta \cos \theta \) to reach the maximum and minimum values of \( \sin \theta + \cos \theta \):
- The maximum value of \( a \cdot \frac{1}{2} \) must be greater than or equal to \( \sqrt{2} \).
- The minimum value of \( a \cdot \frac{1}{2} \) must be less than or equal to \( -\sqrt{2} \).

Thus, we can set up the inequalities:
\[ \frac{a}{2} \geq \sqrt{2} \quad \text{and} \quad \frac{a}{2} \leq -\sqrt{2} \]

### Step 6: Solve the inequalities
From \( \frac{a}{2} \geq \sqrt{2} \):
\[ a \geq 2\sqrt{2} \]
From \( \frac{a}{2} \leq -\sqrt{2} \):
\[ a \leq -2\sqrt{2} \]

### Step 7: Conclusion
Thus, the conditions for \( a \) imply that \( |a| \) must be greater than \( 2\sqrt{2} \). Therefore, the correct choice is:
\[ \text{(a) } |a| > 2\sqrt{2} \]