The condition that the root of ` ax^(2)+bx+c=0` may be in the ratio m: n is :

Options

A. `mnb^(2)=ac(m+n)^(2)`
B. `mnb^(2)=ac(m+n)^(2)`
C. `mna^(2)+bc(m+n)^(2)`
D. `mnc^2=ab(m+n)^(2)`

Answer

To find the condition that the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are in the ratio \( m:n \), we can follow these steps:

### Step-by-Step Solution:

1. **Assume the Roots**:
Let the roots of the equation be \( m\alpha \) and \( n\alpha \), where \( \alpha \) is a common factor.

2. **Sum of Roots**:
According to Vieta's formulas, the sum of the roots is given by:
\[ m\alpha + n\alpha = -\frac{b}{a} \]
This can be simplified to:
\[ \alpha(m + n) = -\frac{b}{a} \]
Therefore, we can express \( \alpha \) as:
\[ \alpha = -\frac{b}{a(m+n)} \]

3. **Product of Roots**:
The product of the roots is given by:
\[ (m\alpha)(n\alpha) = \frac{c}{a} \]
Substituting for \( \alpha \):
\[ mn\alpha^2 = \frac{c}{a} \]
Now substituting the value of \( \alpha \):
\[ mn\left(-\frac{b}{a(m+n)}\right)^2 = \frac{c}{a} \]

4. **Simplifying the Equation**:
Expanding the left side:
\[ mn \cdot \frac{b^2}{a^2(m+n)^2} = \frac{c}{a} \]
Multiplying both sides by \( a^2(m+n)^2 \):
\[ mn b^2 = c a (m+n)^2 \]

5. **Final Condition**:
Rearranging gives us the condition:
\[ b^2mn = ac(m+n)^2 \]

### Conclusion:
The condition that the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are in the ratio \( m:n \) is:
\[ b^2mn = ac(m+n)^2 \]