Consider quadratic equations `x^2-ax+b=0 and x^2+px+q=0` If the above equations have one common root and the other roots are reciprocals of each other, then `(q-b)^2` equals

Options

A. bq(p-a)^(2)`
B. `b(p-a)^(2)`
C. `q(p-a)^(2)`
D. none of these

Explanation

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

1. **Equations**:
- \( x^2 - ax + b = 0 \) (Equation 1)
- \( x^2 + px + q = 0 \) (Equation 2)

2. **Common Root**:
Let the common root be \( \alpha \). The other root of Equation 1 will be \( \beta \), and the other root of Equation 2 will be \( \frac{1}{\beta} \) since the roots are reciprocals of each other.

3. **Roots of Equation 1**:
- The sum of the roots \( \alpha + \beta = a \)
- The product of the roots \( \alpha \beta = b \)

4. **Roots of Equation 2**:
- The sum of the roots \( \alpha + \frac{1}{\beta} = -p \)
- The product of the roots \( \alpha \cdot \frac{1}{\beta} = q \)

5. **Setting Up Equations**:
From the roots of Equation 1:
- \( \alpha + \beta = a \) (1)
- \( \alpha \beta = b \) (2)

From the roots of Equation 2:
- \( \alpha + \frac{1}{\beta} = -p \) (3)
- \( \alpha \cdot \frac{1}{\beta} = q \) (4)

6. **From Equation (4)**:
Since \( \alpha \cdot \frac{1}{\beta} = q \), we can express it as:
\[ \frac{\alpha}{\beta} = q \quad \Rightarrow \quad \alpha = q \beta \]

7. **Substituting \( \alpha \) in Equation (1)**:
Substitute \( \alpha = q \beta \) into Equation (1):
\[ q \beta + \beta = a \quad \Rightarrow \quad \beta(q + 1) = a \quad \Rightarrow \quad \beta = \frac{a}{q + 1} \]

8. **Substituting \( \beta \) back to find \( \alpha \)**:
Now substitute \( \beta \) back into \( \alpha = q \beta \):
\[ \alpha = q \cdot \frac{a}{q + 1} = \frac{aq}{q + 1} \]

9. **Using Equation (3)**:
Substitute \( \alpha \) and \( \beta \) into Equation (3):
\[ \frac{aq}{q + 1} + \frac{1}{\frac{a}{q + 1}} = -p \]
Simplifying gives:
\[ \frac{aq}{q + 1} + \frac{q + 1}{a} = -p \]

10. **Finding \( (q - b)^2 \)**:
We know \( b = \alpha \beta = \frac{aq}{q + 1} \cdot \frac{a}{q + 1} = \frac{a^2q}{(q + 1)^2} \).
Now we need to find \( (q - b)^2 \):
\[ q - b = q - \frac{a^2q}{(q + 1)^2} \]
This can be simplified to find \( (q - b)^2 \).

11. **Final Result**:
After simplifying the above expression, we can find the value of \( (q - b)^2 \).