If tan `theta and cot theta` are roots of `x^(2) + 2ax + b = 0`, then least value of |a| is

Options

A. `(1)/(2)`
B. 1
C. 2
D. cannot be found.

Answer

To solve the problem, we need to analyze the given quadratic equation and the roots provided. The roots of the equation are \( \tan \theta \) and \( \cot \theta \). The quadratic equation is given as:

\[ x^2 + 2ax + b = 0 \]

### Step 1: Identify the roots
The roots of the equation are \( \tan \theta \) and \( \cot \theta \).

### Step 2: Use the properties of roots
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum and product of the roots can be expressed as:
- Sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots \( \alpha \beta = \frac{c}{a} \)

In our case:
- Sum of the roots \( \tan \theta + \cot \theta = -2a \)
- Product of the roots \( \tan \theta \cdot \cot \theta = b \)

### Step 3: Calculate the product of the roots
The product of the roots is:
\[ \tan \theta \cdot \cot \theta = 1 \]
Thus, we have:
\[ b = 1 \]

### Step 4: Calculate the sum of the roots
The sum of the roots can be expressed as:
\[ \tan \theta + \cot \theta = -2a \]
Using the identity \( \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \):
\[ \tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta} = \frac{2}{\sin 2\theta} \]
Thus, we have:
\[ \frac{2}{\sin 2\theta} = -2a \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ a = -\frac{1}{\sin 2\theta} \]

### Step 6: Find the least value of \( |a| \)
To find the least value of \( |a| \):
\[ |a| = \frac{1}{|\sin 2\theta|} \]
The maximum value of \( |\sin 2\theta| \) is 1. Therefore, the minimum value of \( |a| \) is:
\[ |a| = 1 \]

### Conclusion
Thus, the least value of \( |a| \) is:
\[ \boxed{1} \]