The ratio of the roots of the equation `ax^(2)+bx+c=0` is same as the ratio of the 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))/(q^(2))`
B. `(q^(2))/(p^(2))`
C. `(b^(2))/(q^(2))`
D. none of these

Answer

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. **Identify the Roots of the Quadratic Equations**:
The roots of the first quadratic equation \( ax^2 + bx + c = 0 \) can be expressed using the quadratic formula:
\[ r_1 = \frac{-b + \sqrt{D_1}}{2a}, \quad r_2 = \frac{-b - \sqrt{D_1}}{2a} \]
where \( D_1 = b^2 - 4ac \) is the discriminant of the first equation.

Similarly, for the second quadratic equation \( px^2 + qx + r = 0 \):
\[ s_1 = \frac{-q + \sqrt{D_2}}{2p}, \quad s_2 = \frac{-q - \sqrt{D_2}}{2p} \]
where \( D_2 = q^2 - 4pr \) is the discriminant of the second equation.

2. **Set Up the Ratio of the Roots**:
We know that the ratio of the roots of the first equation is equal to the ratio of the roots of the second equation:
\[ \frac{r_1}{r_2} = \frac{s_1}{s_2} \]
Substituting the expressions for \( r_1, r_2, s_1, \) and \( s_2 \):
\[ \frac{\frac{-b + \sqrt{D_1}}{2a}}{\frac{-b - \sqrt{D_1}}{2a}} = \frac{\frac{-q + \sqrt{D_2}}{2p}}{\frac{-q - \sqrt{D_2}}{2p}} \]
This simplifies to:
\[ \frac{-b + \sqrt{D_1}}{-b - \sqrt{D_1}} = \frac{-q + \sqrt{D_2}}{-q - \sqrt{D_2}} \]

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

4. **Expand Both Sides**:
Expanding both sides leads to:
\[ bq + b\sqrt{D_2} - q\sqrt{D_1} - \sqrt{D_1D_2 = bq - b\sqrt{D_2} + q\sqrt{D_1} - \sqrt{D_1D_2} \]
Rearranging gives us:
\[ 2b\sqrt{D_2} = 2q\sqrt{D_1} \]

5. **Simplify the Equation**:
Dividing both sides by 2 gives:
\[ b\sqrt{D_2} = q\sqrt{D_1} \]

6. **Square Both Sides**:
Squaring both sides results in:
\[ b^2D_2 = q^2D_1 \]

7. **Find the Ratio of the Discriminants**:
Rearranging gives us the ratio of the discriminants:
\[ \frac{D_1}{D_2} = \frac{b^2}{q^2} \]

### Final Answer:
Thus, the ratio of the discriminants \( D_1 : D_2 \) is:
\[ D_1 : D_2 = b^2 : q^2 \]