If a and b are the roots of the equations ` P x^(2) - q x + R = 0 ` then what is the value of
`((1)/( a^(2))) + ((1)/( b^(2))) + ((a)/( b)) + ((b)/(a))`

Options

A. `((Q^(2) - 2P) ( 2R + P))/(PR^(2))`
B. `((Q^(2) - 2PR) ( R + P))/(PR^(2))`
C. `((Q^(2) - 2R) ( 2P + R))/(P^(2)R^(2))`
D. `((Q^(2) - 2PR) ( 2R + P))/(P^(2)R^(2))`

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{1}{a^2} + \frac{1}{b^2} + \frac{a}{b} + \frac{b}{a} \]

where \(a\) and \(b\) are the roots of the quadratic equation \(P x^2 - q x + R = 0\).

### Step 1: Use Vieta's Formulas
From Vieta's formulas, we know:
- The sum of the roots \(a + b = \frac{q}{P}\)
- The product of the roots \(ab = \frac{R}{P}\)

### Step 2: Rewrite the Expression
We can rewrite the expression as follows:

\[ \frac{1}{a^2} + \frac{1}{b^2} + \frac{a}{b} + \frac{b}{a} = \frac{b^2 + a^2}{a^2b^2} + \frac{a^2 + b^2}{ab} \]

### Step 3: Find \(a^2 + b^2\)
Using the identity \(a^2 + b^2 = (a + b)^2 - 2ab\):

\[ a^2 + b^2 = \left(\frac{q}{P}\right)^2 - 2\left(\frac{R}{P}\right) = \frac{q^2}{P^2} - \frac{2R}{P} \]

### Step 4: Substitute \(a^2 + b^2\) into the Expression
Now we can substitute \(a^2 + b^2\) back into the expression:

\[ \frac{1}{a^2} + \frac{1}{b^2} + \frac{a}{b} + \frac{b}{a} = \frac{\frac{q^2}{P^2} - \frac{2R}{P}}{ab^2} + \frac{a^2 + b^2}{ab} \]

### Step 5: Simplify the Expression
Substituting \(ab = \frac{R}{P}\):

\[ = \frac{\left(\frac{q^2}{P^2} - \frac{2R}{P}\right)}{\left(\frac{R}{P}\right)^2} + \frac{\left(\frac{q^2}{P^2} - \frac{2R}{P}\right)}{\frac{R}{P}} \]

### Step 6: Combine and Simplify
Now we can simplify the expression further:

1. The first term becomes:

\[ \frac{q^2 - 2RP}{R^2} \]

2. The second term becomes:

\[ \frac{q^2 - 2RP}{R} \]

So we have:

\[ \frac{q^2 - 2RP}{R^2} + \frac{q^2 - 2RP}{R} \]

### Step 7: Find a Common Denominator
Combining these fractions gives:

\[ \frac{(q^2 - 2RP) + (q^2 - 2RP)R}{R^2} = \frac{q^2 - 2RP + q^2R - 2R^2P}{R^2} \]

### Step 8: Final Simplification
This simplifies to:

\[ \frac{q^2(1 + R) - 2P(R + R^2)}{R^2} \]

### Conclusion
After simplifying, we find that the value of the expression is:

\[ \frac{q^2 - 2PR}{PR^2} \]