The ratio of the roots of the equation `ax^2+ bx+c =0` is same as the ratio of roots of equation `px^2+ qx + r =0`. If `D_1 and D_2` are the discriminants of `ax^2+bx +C= 0 and px^2+qx+r=0` respectively, 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 \(D_1\) and \(D_2\) of the two quadratic equations given that the ratio of their roots is the same.

### Step-by-Step Solution:

1. **Write the Quadratic Equations**:
The two quadratic equations are:
\[ ax^2 + bx + c = 0 \quad \text{(1)} \]
\[ px^2 + qx + r = 0 \quad \text{(2)} \]

2. **Identify the Roots**:
Let the roots of equation (1) be \(\alpha_1\) and \(\beta_1\), and the roots of equation (2) be \(\alpha_2\) and \(\beta_2\).

3. **Sum and Product of Roots**:
For equation (1):
- Sum of roots: \(\alpha_1 + \beta_1 = -\frac{b}{a}\)
- Product of roots: \(\alpha_1 \beta_1 = \frac{c}{a}\)

For equation (2):
- Sum of roots: \(\alpha_2 + \beta_2 = -\frac{q}{p}\)
- Product of roots: \(\alpha_2 \beta_2 = \frac{r}{p}\)

4. **Given Condition**:
It is given that the ratio of the roots of the first equation is the same as the ratio of the roots of the second equation:
\[ \frac{\alpha_1}{\beta_1} = \frac{\alpha_2}{\beta_2} \]

5. **Using Compendendo and Dividendo**:
From the given ratio, we can apply the compendendo and dividendo:
\[ \frac{\alpha_1 + \beta_1}{\alpha_1 - \beta_1} = \frac{\alpha_2 + \beta_2}{\alpha_2 - \beta_2} \]

6. **Square Both Sides**:
Squaring both sides gives:
\[ \frac{(\alpha_1 + \beta_1)^2}{(\alpha_1 - \beta_1)^2} = \frac{(\alpha_2 + \beta_2)^2}{(\alpha_2 - \beta_2)^2} \]

7. **Substituting the Values**:
Substitute the sums and products of roots:
\[ \frac{\left(-\frac{b}{a}\right)^2}{\left(-\frac{b}{a}\right)^2 - 4\left(\frac{c}{a}\right)} = \frac{\left(-\frac{q}{p}\right)^2}{\left(-\frac{q}{p}\right)^2 - 4\left(\frac{r}{p}\right)} \]

8. **Simplifying**:
This simplifies to:
\[ \frac{\frac{b^2}{a^2}}{\frac{b^2}{a^2} - \frac{4c}{a}} = \frac{\frac{q^2}{p^2}}{\frac{q^2}{p^2} - \frac{4r}{p}} \]

9. **Cross-Multiplying**:
Cross-multiplying gives:
\[ b^2 \left(\frac{q^2}{p^2} - 4r/p\right) = q^2 \left(\frac{b^2}{a^2} - 4c/a\right) \]

10. **Identifying Discriminants**:
The discriminants are:
\[ D_1 = b^2 - 4ac \quad \text{and} \quad D_2 = q^2 - 4pr \]

11. **Final Ratio**:
From the previous steps, we can derive:
\[ \frac{D_1}{D_2} = \frac{b^2}{q^2} \]
Therefore, the ratio of the discriminants is:
\[ D_1 : D_2 = b^2 : q^2 \]

### Final Answer:
The required ratio is \(D_1 : D_2 = b^2 : q^2\).