let `alpha ,beta` be roots of `ax^2+bx+c=0` and `gamma,delta` be the roots of `px^2+qx+r=0`and `D_1` and `D_2` be the respective equations .if `alpha,beta,gamma,delta` in `A.P.` then `D_1/D_2` is

Options

A. `a^(2)/b^(2)`
B. `a^(2)/p^(2)`
C. `b^(2)/q^(2)`
D. `c^(2)/r^(2)`

Answer

To solve the problem, we need to find the ratio \( \frac{D_1}{D_2} \) where \( D_1 \) and \( D_2 \) are the discriminants of the quadratic equations given the roots are in arithmetic progression (A.P.).

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots \( \alpha, \beta, \gamma, \delta \) be in A.P. We can express them as:
- \( \alpha = a \)
- \( \beta = a + d \)
- \( \gamma = a + 2d \)
- \( \delta = a + 3d \)

2. **Sum and Product of Roots for the First Equation**:
For the equation \( ax^2 + bx + c = 0 \):
- The sum of roots \( \alpha + \beta = a + (a + d) = 2a + d = -\frac{b}{a} \)
- The product of roots \( \alpha \beta = a(a + d) = c/a \)

3. **Sum and Product of Roots for the Second Equation**:
For the equation \( px^2 + qx + r = 0 \):
- The sum of roots \( \gamma + \delta = (a + 2d) + (a + 3d) = 2a + 5d = -\frac{q}{p} \)
- The product of roots \( \gamma \delta = (a + 2d)(a + 3d) = r/p \)

4. **Expressing the Discriminants**:
The discriminants \( D_1 \) and \( D_2 \) are given by:
- \( D_1 = b^2 - 4ac \)
- \( D_2 = q^2 - 4pr \)

5. **Calculating \( D_1 \)**:
Using the relations from the roots:
- From \( 2a + d = -\frac{b}{a} \), we can express \( b \):
\[ b = -a(2a + d) \]
- The product gives:
\[ c = a(a + d) \implies c = a^2 + ad \]
- Thus, substituting into \( D_1 \):
\[ D_1 = \left(-a(2a + d)\right)^2 - 4a(a^2 + ad) \]
\[ = a^2(2a + d)^2 - 4a(a^2 + ad) \]

6. **Calculating \( D_2 \)**:
Similarly for \( D_2 \):
- From \( 2a + 5d = -\frac{q}{p} \), we can express \( q \):
\[ q = -p(2a + 5d) \]
- The product gives:
\[ r = p(a + 2d)(a + 3d) \]
- Thus, substituting into \( D_2 \):
\[ D_2 = \left(-p(2a + 5d)\right)^2 - 4p(a + 2d)(a + 3d) \]
\[ = p^2(2a + 5d)^2 - 4p(a + 2d)(a + 3d) \]

7. **Finding the Ratio \( \frac{D_1}{D_2} \)**:
Finally, the ratio \( \frac{D_1}{D_2} \) can be simplified:
\[ \frac{D_1}{D_2} = \frac{a^2(2a + d)^2 - 4a(a^2 + ad)}{p^2(2a + 5d)^2 - 4p(a + 2d)(a + 3d)} \]

### Final Result:
The final answer for \( \frac{D_1}{D_2} \) is:
\[ \frac{D_1}{D_2} = \frac{a^2}{p^2} \]