The ration of roots of the equations `ax^(2) + bx + c =0` and px^(2) + qx + r = 0` " are equal. If " D_(1) and D_(2) ` are respective discriminates. Then what is `(D_(1))/(D_(2))` equal to ?

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 equal.

### Step-by-Step Solution:

1. **Understand 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. **Roots of the Quadratic Equations**:
The roots of a quadratic equation \( Ax^2 + Bx + C = 0 \) can be found using the quadratic formula:
\[ x = \frac{-B \pm \sqrt{D}}{2A} \]
where \( D \) is the discriminant given by:
\[ D = B^2 - 4AC \]

3. **Discriminants**:
For the first equation (1):
\[ D_1 = b^2 - 4ac \]
For the second equation (2):
\[ D_2 = q^2 - 4pr \]

4. **Ratio of Roots**:
The ratio of the roots of the first equation is given by:
\[ \text{Ratio}_1 = \frac{-b + \sqrt{D_1}}{-b - \sqrt{D_1}} = \frac{b + \sqrt{D_1}}{b - \sqrt{D_1}} \]
The ratio of the roots of the second equation is:
\[ \text{Ratio}_2 = \frac{-q + \sqrt{D_2}}{-q - \sqrt{D_2}} = \frac{q + \sqrt{D_2}}{q - \sqrt{D_2}} \]

5. **Setting the Ratios Equal**:
Since the problem states that the ratios of the roots are equal, we have:
\[ \frac{b + \sqrt{D_1}}{b - \sqrt{D_1}} = \frac{q + \sqrt{D_2}}{q - \sqrt{D_2}} \]

6. **Cross Multiplying**:
Cross-multiplying gives us:
\[ (b + \sqrt{D_1})(q - \sqrt{D_2}) = (q + \sqrt{D_2})(b - \sqrt{D_1}) \]

7. **Simplifying**:
By simplifying the above equation, we can derive a relationship between \( D_1 \) and \( D_2 \).

8. **Finding the Ratio of Discriminants**:
After simplification, we find that:
\[ \frac{D_1}{D_2} = \frac{b^2}{q^2} \]

### Final Result:
Thus, the ratio of the discriminants \( \frac{D_1}{D_2} \) is equal to:
\[ \frac{D_1}{D_2} = \frac{b^2}{q^2} \]