If the ratio of the roots of `x^(2) + bx + c = 0` is equal to the ratio of the roots of `x^(2)+px+q=0`, then `p^(2)c-b^(2)q=`

Options

A. a) 0
B. b) 1
C. c) 2
D. d) 3

Answer

To solve the problem, we need to find the value of \( p^2c - b^2q \) given that the ratio of the roots of the equations \( x^2 + bx + c = 0 \) and \( x^2 + px + q = 0 \) are equal.

### Step-by-step Solution:

1. **Identify the Roots**:
- For the equation \( x^2 + bx + c = 0 \), let the roots be \( \alpha \) and \( \beta \).
- The sum of the roots \( \alpha + \beta = -b \).
- The product of the roots \( \alpha \beta = c \).

- For the equation \( x^2 + px + q = 0 \), let the roots be \( \gamma \) and \( \delta \).
- The sum of the roots \( \gamma + \delta = -p \).
- The product of the roots \( \gamma \delta = q \).

2. **Set Up the Ratio of Roots**:
- We are given that the ratio of the roots of the first equation is equal to the ratio of the roots of the second equation:
\[ \frac{\alpha}{\beta} = \frac{\gamma}{\delta} \]

3. **Apply Componendo-Dividendo**:
- By applying the Componendo-Dividendo method, we can rewrite the ratio:
\[ \frac{\alpha + \beta}{\alpha - \beta} = \frac{\gamma + \delta}{\gamma - \delta} \]

4. **Substitute the Values**:
- Substitute the values of the sums and differences of the roots:
\[ \frac{-b}{\alpha - \beta} = \frac{-p}{\gamma - \delta} \]

5. **Square Both Sides**:
- Squaring both sides gives us:
\[ \left(\frac{-b}{\alpha - \beta}\right)^2 = \left(\frac{-p}{\gamma - \delta}\right)^2 \]

6. **Express Differences**:
- The differences \( \alpha - \beta \) and \( \gamma - \delta \) can be expressed using the square root of the discriminants:
\[ \alpha - \beta = \sqrt{b^2 - 4c}, \quad \gamma - \delta = \sqrt{p^2 - 4q} \]

7. **Substituting Back**:
- Now substituting these into the squared ratio:
\[ \frac{b^2}{b^2 - 4c} = \frac{p^2}{p^2 - 4q} \]

8. **Cross Multiply**:
- Cross multiplying gives:
\[ b^2(p^2 - 4q) = p^2(b^2 - 4c) \]

9. **Rearranging the Equation**:
- Rearranging the equation leads to:
\[ p^2b^2 - 4b^2q = p^2b^2 - 4p^2c \]

10. **Simplifying**:
- Cancel \( p^2b^2 \) from both sides:
\[ -4b^2q = -4p^2c \]

11. **Final Result**:
- Dividing by -4 gives:
\[ p^2c - b^2q = 0 \]

### Conclusion:
Thus, the value of \( p^2c - b^2q \) is \( 0 \).