If the roots of the equation `ax^(2)+bx+c=0` are in the ratio `m:n` then

Options

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

Answer

To solve the problem, we start with the given quadratic equation:

\[ ax^2 + bx + c = 0 \]

We know that if the roots of this equation are in the ratio \( m:n \), we can express the roots as:

\[ \alpha = mx \quad \text{and} \quad \beta = nx \]

where \( x \) is a common factor.

### Step 1: Write the sum of the roots
The sum of the roots \( \alpha + \beta \) can be expressed as:

\[ \alpha + \beta = mx + nx = (m+n)x \]

According to Vieta's formulas, the sum of the roots for the quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ \alpha + \beta = -\frac{b}{a} \]

### Step 2: Set the equations equal
Now we can set the two expressions for the sum of the roots equal to each other:

\[ (m+n)x = -\frac{b}{a} \]

### Step 3: Write the product of the roots
Next, we find the product of the roots \( \alpha \beta \):

\[ \alpha \beta = (mx)(nx) = mnx^2 \]

According to Vieta's formulas, the product of the roots for the quadratic equation is given by:

\[ \alpha \beta = \frac{c}{a} \]

### Step 4: Set the equations equal
Now we can set the two expressions for the product of the roots equal to each other:

\[ mnx^2 = \frac{c}{a} \]

### Step 5: Solve for \( x \)
From the first equation, we can solve for \( x \):

\[ x = -\frac{b}{a(m+n)} \]

Substituting this value of \( x \) into the product equation:

\[ mn\left(-\frac{b}{a(m+n)}\right)^2 = \frac{c}{a} \]

### Step 6: Simplify the equation
Now we simplify the equation:

\[ mn\frac{b^2}{a^2(m+n)^2} = \frac{c}{a} \]

Multiplying both sides by \( a^2(m+n)^2 \):

\[ mn b^2 = c a (m+n)^2 \]

### Conclusion
Thus, we have established a relationship between the coefficients \( a, b, c \) and the ratios \( m, n \):

\[ mn b^2 = c a (m+n)^2 \]