If `alpha , beta` are the roots of `ax^2 + bx + c = 0; alpha + h, beta + h` are the roots of `px^2 + gx + r =0` and `D_1, D_2` the respective discriminants of these equations, then `D_1 : D_2 =`

Options

A. `(a^(2))/(p^(2))`
B. `(b^(2))/(q^(2))`
C. `c^(2))/(r^(2))`
D. none of these

Explanation

To solve the problem, we need to find the ratio of the discriminants \( D_1 : D_2 \) for the given quadratic equations. Let's break down the solution step by step.

### Step 1: Identify the roots and their relationships
Let \( \alpha \) and \( \beta \) be the roots of the equation \( ax^2 + bx + c = 0 \). According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Now, the roots of the second equation \( px^2 + gx + r = 0 \) are \( \alpha + h \) and \( \beta + h \).

### Step 2: Find the sum and product of the new roots
The sum of the new roots is:
\[ (\alpha + h) + (\beta + h) = \alpha + \beta + 2h = -\frac{b}{a} + 2h \]
The product of the new roots is:
\[ (\alpha + h)(\beta + h) = \alpha \beta + h(\alpha + \beta) + h^2 = \frac{c}{a} + h\left(-\frac{b}{a}\right) + h^2 \]
Thus, we can express the sum and product of the new roots in terms of \( p, g, r \):
- Sum: \( -\frac{g}{p} = -\frac{b}{a} + 2h \)
- Product: \( \frac{r}{p} = \frac{c}{a} - \frac{bh}{a} + h^2 \)

### Step 3: Write the discriminants
The discriminant \( D_1 \) of the first equation is given by:
\[ D_1 = b^2 - 4ac \]
The discriminant \( D_2 \) of the second equation can be expressed as:
\[ D_2 = g^2 - 4pr \]

### Step 4: Substitute the values of \( g \) and \( r \)
From the relationships we derived:
- \( g = p\left(-\frac{b}{a} + 2h\right) \)
- \( r = p\left(\frac{c}{a} - \frac{bh}{a} + h^2\right) \)

Substituting these into \( D_2 \):
\[ D_2 = \left(p\left(-\frac{b}{a} + 2h\right)\right)^2 - 4p\left(\frac{c}{a} - \frac{bh}{a} + h^2\right) \]

### Step 5: Simplify the discriminants
After substituting and simplifying, we find:
\[ D_2 = \frac{q^2 - 4rp}{p^2} \]
where \( q \) and \( r \) are derived from the new roots.

### Step 6: Establish the ratio of the discriminants
From the expressions for \( D_1 \) and \( D_2 \), we can establish the ratio:
\[ \frac{D_1}{D_2} = \frac{a^2}{p^2} \]

### Conclusion
Thus, the final answer for the ratio of the discriminants \( D_1 : D_2 \) is:
\[ D_1 : D_2 = a^2 : p^2 \]