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 \( ax^2 + bx + c = 0 \) may be in the ratio \( m:n \), we can follow these steps:

### Step 1: Define the Roots
Let the roots of the equation be \( mx \) and \( nx \), where \( m \) and \( n \) are the given ratios.

### Step 2: Use Vieta's Formulas
According to Vieta's formulas, for the quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( (mx + nx) \) is equal to \( -\frac{b}{a} \).
- The product of the roots \( (mx)(nx) \) is equal to \( \frac{c}{a} \).

### Step 3: Express the Sum of the Roots
From the sum of the roots:
\[ mx + nx = (m+n)x = -\frac{b}{a} \]
This gives us:
\[ x = -\frac{b}{a(m+n)} \]

### Step 4: Express the Product of the Roots
From the product of the roots:
\[ (mx)(nx) = mnx^2 = \frac{c}{a} \]
Substituting \( x \) from the previous step:
\[ mn\left(-\frac{b}{a(m+n)}\right)^2 = \frac{c}{a} \]

### Step 5: Simplify the Equation
Squaring \( x \):
\[ mn\left(\frac{b^2}{a^2(m+n)^2}\right) = \frac{c}{a} \]
Multiplying both sides by \( a^2(m+n)^2 \):
\[ mnb^2 = ac(m+n)^2 \]

### Step 6: Final Condition
Rearranging gives us the final condition:
\[ ac(m+n)^2 = b^2mn \]

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