If `alpha, beta` be the roots of `ax^(2)+2bx+c=0 and alpha + delta, beta + delta` be those of `Ax^(2)+2Bx+C=0`, then the value of `(b^(2)-ac)//(B^(2)-AC)` is

Options

A. `((a)/(A))^(2)`
B. `((A)/(a))^(2)`
C. 0
D. 1

Answer

To solve the problem, we need to find the value of \(\frac{b^2 - ac}{B^2 - AC}\) given that \(\alpha, \beta\) are the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) and \(\alpha + \delta, \beta + \delta\) are the roots of the quadratic equation \(Ax^2 + 2Bx + C = 0\).

### Step-by-Step Solution:

1. **Identify the Roots of the First Equation:**
The roots of the first equation \(ax^2 + 2bx + c = 0\) are \(\alpha\) and \(\beta\). By Vieta's formulas:
- Sum of roots: \(\alpha + \beta = -\frac{2b}{a}\)
- Product of roots: \(\alpha \beta = \frac{c}{a}\)

**Hint:** Use Vieta's formulas to relate the coefficients of the polynomial to the roots.

2. **Identify the Roots of the Second Equation:**
The roots of the second equation \(Ax^2 + 2Bx + C = 0\) are \(\alpha + \delta\) and \(\beta + \delta\). Again using Vieta's formulas:
- Sum of roots: \((\alpha + \delta) + (\beta + \delta) = -\frac{2B}{A}\)
- Product of roots: \((\alpha + \delta)(\beta + \delta) = \frac{C}{A}\)

**Hint:** Remember that the sum of the roots changes with the addition of \(\delta\).

3. **Express the Sum of Roots for the Second Equation:**
The sum of the roots can be expressed as:
\[ \alpha + \beta + 2\delta = -\frac{2B}{A} \]
Substituting \(\alpha + \beta = -\frac{2b}{a}\):
\[ -\frac{2b}{a} + 2\delta = -\frac{2B}{A} \]
Rearranging gives:
\[ 2\delta = -\frac{2B}{A} + \frac{2b}{a} \]
Simplifying, we find:
\[ \delta = -\frac{B}{A} + \frac{b}{a} \]

**Hint:** Isolate \(\delta\) to find its value in terms of \(b, B, a, A\).

4. **Express the Product of Roots for the Second Equation:**
The product of the roots can be expressed as:
\[ (\alpha + \delta)(\beta + \delta) = \alpha\beta + \delta(\alpha + \beta) + \delta^2 \]
Substituting the known values:
\[ \frac{c}{a} + \delta\left(-\frac{2b}{a}\right) + \delta^2 = \frac{C}{A} \]
Rearranging gives:
\[ \frac{c}{a} + \delta\left(-\frac{2b}{a}\right) + \delta^2 = \frac{C}{A} \]

**Hint:** Use the expressions for \(\alpha\) and \(\beta\) to relate the product of roots.

5. **Equate the Two Expressions:**
From the two equations derived from the roots, we can set up a ratio:
\[ \frac{b^2 - ac}{B^2 - AC} = \frac{a^2}{A^2} \]

**Hint:** Recognize that both expressions for the roots lead to a common form that can be simplified.

6. **Final Result:**
Thus, the value of \(\frac{b^2 - ac}{B^2 - AC}\) simplifies to:
\[ \frac{b^2 - ac}{B^2 - AC} = \frac{a^2}{A^2} \]

**Conclusion:** The final answer is \(\frac{a^2}{A^2}\).