If `tanalpha,tan beta` are roots of the equation `(1 + cos theta) tan^(2) x - lambda tan x = 1 - cos theta` where `theta in (0, (pi)/(2))` and `cos^(2) (alpha + beta) = (1)/(51)` then `lambda` is :

Options

A. `5sqrt(2)`
B. `10 sqrt(2)`
C. 10
D. 5

Explanation

To solve the given problem, we will follow a systematic approach.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The given equation is:
\[ (1 + \cos \theta) \tan^2 x - \lambda \tan x = 1 - \cos \theta \]
This is a quadratic equation in terms of \(\tan x\).

2. **Identifying Roots**:
We know that \(\tan \alpha\) and \(\tan \beta\) are the roots of this quadratic equation. By Vieta's formulas, we can express the sum and product of the roots:
- Sum of the roots:
\[ \tan \alpha + \tan \beta = \frac{\lambda}{1 + \cos \theta} \]
- Product of the roots:
\[ \tan \alpha \tan \beta = \frac{1 - \cos \theta}{1 + \cos \theta} \]

3. **Using the Cosine Identity**:
We are given:
\[ \cos^2(\alpha + \beta) = \frac{1}{51} \]
Using the identity \(\cos^2(\alpha + \beta) = \frac{1}{1 + \tan^2(\alpha + \beta)}\), we can write:
\[ 51 = 1 + \tan^2(\alpha + \beta) \]
Therefore:
\[ \tan^2(\alpha + \beta) = 50 \]

4. **Expressing \(\tan(\alpha + \beta)\)**:
We know that:
\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]
Substituting the values from Vieta's formulas:
\[ \tan(\alpha + \beta) = \frac{\frac{\lambda}{1 + \cos \theta}}{1 - \frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\frac{\lambda}{1 + \cos \theta}}{\frac{2\cos \theta}{1 + \cos \theta}} = \frac{\lambda}{2\cos \theta} \]

5. **Setting Up the Equation**:
Since we have \(\tan^2(\alpha + \beta) = 50\), we can equate:
\[ \left(\frac{\lambda}{2\cos \theta}\right)^2 = 50 \]

6. **Solving for \(\lambda\)**:
Squaring both sides gives:
\[ \frac{\lambda^2}{4\cos^2 \theta} = 50 \]
Rearranging gives:
\[ \lambda^2 = 200\cos^2 \theta \]

7. **Finding \(\lambda\)**:
Taking the square root:
\[ \lambda = \sqrt{200\cos^2 \theta} = 10\sqrt{2}|\cos \theta| \]
Since \(\theta\) is in \((0, \frac{\pi}{2})\), \(\cos \theta\) is positive, thus:
\[ \lambda = 10\sqrt{2}\cos \theta \]

### Final Answer:
The value of \(\lambda\) is \(10\sqrt{2}\cos \theta\).