Let `alpha,beta` be the roots of the equation `x^(2)-px+r=0 and alpha//2,2beta` be the roots of the equation `x^(2)-qx+r=0`, then the value of r is (1)`(2)/(9)(p-q)(2q-p)` (2) `(2)/(9)(q-p)(2p-q)` (3)`(2)/(9)(q-2p)(2q-p)` (4)`(2)/(9)(2p-q)(2q-p)`

Options

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

Answer

To solve the problem, we start with the two quadratic equations given:

1. The first equation is \(x^2 - px + r = 0\) with roots \(\alpha\) and \(\beta\).
2. The second equation is \(x^2 - qx + r = 0\) with roots \(\frac{\alpha}{2}\) and \(2\beta\).

### Step 1: Use Vieta's Formulas for the First Equation
From the first equation, by Vieta's formulas, we know:
- The sum of the roots:
\[ \alpha + \beta = p \]
- The product of the roots:
\[ \alpha \beta = r \]

### Step 2: Use Vieta's Formulas for the Second Equation
For the second equation, the roots are \(\frac{\alpha}{2}\) and \(2\beta\). Again, using Vieta's formulas:
- The sum of the roots:
\[ \frac{\alpha}{2} + 2\beta = q \]
- The product of the roots:
\[ \left(\frac{\alpha}{2}\right)(2\beta) = r \]

### Step 3: Simplify the Sum of the Roots for the Second Equation
We can rewrite the sum of the roots from the second equation:
\[ \frac{\alpha}{2} + 2\beta = q \]
Multiplying through by 2 to eliminate the fraction gives:
\[ \alpha + 4\beta = 2q \]

### Step 4: Substitute \(\alpha + \beta\) into the Equation
From the first equation, we know \(\alpha + \beta = p\). We can express \(\alpha\) in terms of \(\beta\):
\[ \alpha = p - \beta \]
Substituting this into the equation \(\alpha + 4\beta = 2q\):
\[ (p - \beta) + 4\beta = 2q \]
This simplifies to:
\[ p + 3\beta = 2q \]
Rearranging gives:
\[ 3\beta = 2q - p \quad \Rightarrow \quad \beta = \frac{2q - p}{3} \]

### Step 5: Find \(\alpha\)
Now substituting \(\beta\) back to find \(\alpha\):
\[ \alpha = p - \beta = p - \frac{2q - p}{3} \]
This simplifies to:
\[ \alpha = \frac{3p - (2q - p)}{3} = \frac{4p - 2q}{3} \]

### Step 6: Calculate \(r\)
Now we have \(\alpha\) and \(\beta\). We can find \(r\) using the product of the roots:
\[ r = \alpha \beta = \left(\frac{4p - 2q}{3}\right) \left(\frac{2q - p}{3}\right) \]
Calculating this gives:
\[ r = \frac{(4p - 2q)(2q - p)}{9} \]
Expanding this:
\[ = \frac{8pq - 4p^2 - 4q^2 + 2pq}{9} = \frac{10pq - 4p^2 - 4q^2}{9} \]

### Step 7: Final Form of \(r\)
We can express \(r\) in terms of the options given. The expression can be factored to match one of the options:
\[ r = \frac{2}{9}(2p - q)(2q - p) \]

### Conclusion
Thus, the value of \(r\) is:
\[ \boxed{\frac{2}{9}(2p - q)(2q - p)} \]