If `alpha` and `beta` are roots of `x^2+px+2=0` and `1/alpha,1/beta` are the roots of `2x^2+2qx+1=0`. Find the value of `(alpha-1/alpha)(beta-1/beta)(alpha+1/beta)(beta+1/alpha)`

Options

A. `9/4(9-p^2)`
B. `9/4(9+q^2)`
C. `9/4(9+p^2)`
D. `9/4(9-q^2)`

Answer

To solve the problem, we need to find the value of the expression \((\alpha - \frac{1}{\alpha})(\beta - \frac{1}{\beta})(\alpha + \frac{1}{\beta})(\beta + \frac{1}{\alpha})\) given that \(\alpha\) and \(\beta\) are roots of the equation \(x^2 + px + 2 = 0\) and \(\frac{1}{\alpha}, \frac{1}{\beta}\) are roots of the equation \(2x^2 + 2qx + 1 = 0\).

### Step 1: Find \(\alpha + \beta\) and \(\alpha \beta\)
From the equation \(x^2 + px + 2 = 0\), we can use Vieta's formulas:
- The sum of the roots \(\alpha + \beta = -p\)
- The product of the roots \(\alpha \beta = 2\)

### Step 2: Find \(\frac{1}{\alpha} + \frac{1}{\beta}\) and \(\frac{1}{\alpha} \cdot \frac{1}{\beta}\)
For the equation \(2x^2 + 2qx + 1 = 0\):
- The sum of the roots \(\frac{1}{\alpha} + \frac{1}{\beta} = -\frac{2q}{2} = -q\)
- The product of the roots \(\frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{2}\)

Using the relationship for the sum of reciprocals:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \]
Substituting the values we found:
\[ -q = \frac{-p}{2} \implies p = 2q \]

### Step 3: Substitute values into the expression
The expression we need to evaluate is:
\[ (\alpha - \frac{1}{\alpha})(\beta - \frac{1}{\beta})(\alpha + \frac{1}{\beta})(\beta + \frac{1}{\alpha}) \]

We can rewrite each term:
1. \(\alpha - \frac{1}{\alpha} = \frac{\alpha^2 - 1}{\alpha}\)
2. \(\beta - \frac{1}{\beta} = \frac{\beta^2 - 1}{\beta}\)
3. \(\alpha + \frac{1}{\beta} = \frac{\alpha \beta + 1}{\beta}\)
4. \(\beta + \frac{1}{\alpha} = \frac{\alpha \beta + 1}{\alpha}\)

Thus, the expression becomes:
\[ \frac{(\alpha^2 - 1)(\beta^2 - 1)(\alpha \beta + 1)^2}{\alpha \beta^2 \cdot \alpha^2 \beta} \]

### Step 4: Substitute \(\alpha \beta\) and simplify
Substituting \(\alpha \beta = 2\):
\[ = \frac{(\alpha^2 - 1)(\beta^2 - 1)(2 + 1)^2}{2^2} \]
\[ = \frac{(\alpha^2 - 1)(\beta^2 - 1) \cdot 9}{4} \]

### Step 5: Find \(\alpha^2 + \beta^2\)
Using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \]
Substituting the values:
\[ = (-p)^2 - 2 \cdot 2 = p^2 - 4 \]

### Step 6: Substitute \(\alpha^2 + \beta^2\) into the expression
Now we can express \((\alpha^2 - 1)(\beta^2 - 1)\):
\[ (\alpha^2 - 1)(\beta^2 - 1) = \alpha^2 \beta^2 - (\alpha^2 + \beta^2) + 1 \]
Substituting \(\alpha^2 \beta^2 = (\alpha \beta)^2 = 4\):
\[ = 4 - (p^2 - 4) + 1 = 9 - p^2 \]

### Final Expression
Putting it all together:
\[ = \frac{9(9 - p^2)}{4} \]

Thus, the final value of the expression is:
\[ \frac{9(9 - p^2)}{4} \]