If the quadratic equation `x^(2)-36x+lambda=0` has roots `alpha` and `beta` such that `alpha`, `beta in N` and `(lambda)/(5) cancel in Z` and `lambda` assumes minimum possible value then `(sqrt(alpha+2)sqrt(beta+2))/(|alpha-beta|)` is equal to (a) `(3)/(8)` (b) `(3)/(16)` (c) `(sqrt(111))/(34)` (d) `(sqrt(111))/(17)`

Options

A. `(3)/(8)`
B. `(3)/(16)`
C. `(sqrt(111))/(34)`
D. `(sqrt(111))/(17)`

Explanation

To solve the given problem, we start with the quadratic equation:

\[ x^2 - 36x + \lambda = 0 \]

### Step 1: Identify the roots
Let the roots of the equation be \( \alpha \) and \( \beta \). According to Vieta's formulas, we have:

- \( \alpha + \beta = 36 \)
- \( \alpha \beta = \lambda \)

### Step 2: Determine conditions for \( \lambda \)
We know that \( \lambda \) must be a natural number and \( \frac{\lambda}{5} \) should not be an integer, meaning \( \lambda \) is not divisible by 5. We want to find the minimum possible value of \( \lambda \).

### Step 3: Explore possible pairs of \( \alpha \) and \( \beta \)
Since \( \alpha + \beta = 36 \), we can express \( \beta \) in terms of \( \alpha \):

\[ \beta = 36 - \alpha \]

Now, substituting this into the equation for \( \lambda \):

\[ \lambda = \alpha(36 - \alpha) = 36\alpha - \alpha^2 \]

### Step 4: Check values of \( \alpha \)
We will check values of \( \alpha \) starting from 1 up to 35 (since both \( \alpha \) and \( \beta \) must be natural numbers):

1. **For \( \alpha = 1 \)**:
- \( \beta = 35 \)
- \( \lambda = 1 \times 35 = 35 \) (divisible by 5, not valid)

2. **For \( \alpha = 2 \)**:
- \( \beta = 34 \)
- \( \lambda = 2 \times 34 = 68 \) (not divisible by 5, valid)

3. **For \( \alpha = 3 \)**:
- \( \beta = 33 \)
- \( \lambda = 3 \times 33 = 99 \) (not divisible by 5, valid)

4. **For \( \alpha = 4 \)**:
- \( \beta = 32 \)
- \( \lambda = 4 \times 32 = 128 \) (not divisible by 5, valid)

Continuing this way, we find that \( \lambda = 68 \) is the minimum valid value that satisfies all conditions.

### Step 5: Calculate the expression
Now we need to calculate:

\[ \frac{\sqrt{\alpha + 2} \cdot \sqrt{\beta + 2}}{|\alpha - \beta|} \]

Substituting \( \alpha = 2 \) and \( \beta = 34 \):

\[ \sqrt{\alpha + 2} = \sqrt{2 + 2} = \sqrt{4} = 2 \]

\[ \sqrt{\beta + 2} = \sqrt{34 + 2} = \sqrt{36} = 6 \]

Now, substituting these values into the expression:

\[ \frac{2 \cdot 6}{|2 - 34|} = \frac{12}{32} = \frac{3}{8} \]

### Final Answer
Thus, the value of the expression is:

\[ \frac{3}{8} \]