If `alpha , beta ` are the roots of the equation ` x^(2)-px+q=0`, then ` (alpha^(2))/(beta^(2))+(beta^(2))/(alpha^(2))` is equal to :

Options

A. `(p^4)/(q^(2))+2-(4p^(2))/(q)`
B. `(p^4)/(q^(2))-2+(4p^(2))/(q)`
C. `(p^(4))/(q^(2))+2q^(2)-(4p^(2))/(q)`
D. None of these

Answer

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

### Step-by-Step Solution:

1. **Identify the Roots**:
The roots of the quadratic equation \(x^2 - px + q = 0\) are \(\alpha\) and \(\beta\). According to Vieta's formulas:
\[ \alpha + \beta = p \quad \text{(sum of roots)} \]
\[ \alpha \beta = q \quad \text{(product of roots)} \]

2. **Rewrite the Expression**:
We need to simplify the expression:
\[ \frac{\alpha^2}{\beta^2} + \frac{\beta^2}{\alpha^2} \]
This can be rewritten as:
\[ \frac{\alpha^4 + \beta^4}{\alpha^2 \beta^2} \]

3. **Find \(\alpha^4 + \beta^4\)**:
To find \(\alpha^4 + \beta^4\), we can use the identity:
\[ \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2\alpha^2\beta^2 \]

4. **Calculate \(\alpha^2 + \beta^2\)**:
First, we find \(\alpha^2 + \beta^2\) using the square of the sum of roots:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = p^2 - 2q \]

5. **Substitute into the Identity**:
Now substitute \(\alpha^2 + \beta^2\) into the identity for \(\alpha^4 + \beta^4\):
\[ \alpha^4 + \beta^4 = (p^2 - 2q)^2 - 2q^2 \]

6. **Expand and Simplify**:
Expanding \((p^2 - 2q)^2\):
\[ (p^2 - 2q)^2 = p^4 - 4pq + 4q^2 \]
Therefore,
\[ \alpha^4 + \beta^4 = p^4 - 4pq + 4q^2 - 2q^2 = p^4 - 4pq + 2q^2 \]

7. **Substitute Back into the Expression**:
Now substitute \(\alpha^4 + \beta^4\) back into the expression we need:
\[ \frac{\alpha^4 + \beta^4}{\alpha^2 \beta^2} = \frac{p^4 - 4pq + 2q^2}{q^2} \]

8. **Final Simplification**:
This simplifies to:
\[ \frac{p^4}{q^2} - \frac{4p}{q} + 2 \]

### Final Answer:
Thus, the value of \(\frac{\alpha^2}{\beta^2} + \frac{\beta^2}{\alpha^2}\) is:
\[ \frac{p^4}{q^2} - \frac{4p}{q} + 2 \]