If the roots of the equation `x^(2)-5x+16=0` are `alpha, beta` and the roots of the equation `x^(2)+ax+b=0` are `alpha^(2)+beta^(2) and (alpha beta)/(2)`, then (a,b) is

Options

A. (1,56)
B. (1,-56)
C. (-1,56)
D. (-1,-56)

Answer

To solve the problem, we will follow these steps:

1. **Identify the roots of the first equation**: The given equation is \(x^2 - 5x + 16 = 0\). The roots of this equation are denoted as \(\alpha\) and \(\beta\).

2. **Calculate the sum and product of the roots**:
- The sum of the roots \(\alpha + \beta\) is given by the formula \(-\frac{b}{a}\), where \(b\) is the coefficient of \(x\) and \(a\) is the coefficient of \(x^2\).
- Here, \(a = 1\) and \(b = -5\), so:
\[ \alpha + \beta = -\frac{-5}{1} = 5 \]
- The product of the roots \(\alpha \beta\) is given by \(\frac{c}{a}\), where \(c\) is the constant term.
- Here, \(c = 16\), so:
\[ \alpha \beta = \frac{16}{1} = 16 \]

3. **Determine the roots of the second equation**: The roots of the second equation \(x^2 + ax + b = 0\) are given as \(\alpha^2 + \beta^2\) and \(\frac{\alpha \beta}{2}\).

4. **Calculate \(\alpha^2 + \beta^2\)**:
- We can use the identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\).
- Substituting the values we found:
\[ \alpha^2 + \beta^2 = (5)^2 - 2(16) = 25 - 32 = -7 \]

5. **Calculate \(\frac{\alpha \beta}{2}\)**:
- We already found \(\alpha \beta = 16\), so:
\[ \frac{\alpha \beta}{2} = \frac{16}{2} = 8 \]

6. **Set up the equations for the second quadratic**:
- The sum of the roots of the second equation is:
\[ \alpha^2 + \beta^2 + \frac{\alpha \beta}{2} = -7 + 8 = 1 \]
- According to Vieta's formulas, the sum of the roots is equal to \(-a\):
\[ -a = 1 \implies a = -1 \]

7. **Calculate the product of the roots for the second equation**:
- The product of the roots is:
\[ \left(\alpha^2 + \beta^2\right) \left(\frac{\alpha \beta}{2}\right) = (-7)(8) = -56 \]
- According to Vieta's formulas, the product of the roots is equal to \(b\):
\[ b = -56 \]

8. **Final answer**: Therefore, the values of \(a\) and \(b\) are:
\[ (a, b) = (-1, -56) \]