If r be the ratio of the roots of the equation `ax^2+bx+c=0`, then which one of the following is true

Options

A. `ac(1+r)^2=(rb)^2`
B. `ac(1+r)^2=rb^2`
C. `ab(1+r)^2=rc^2`
D. `ac(1+r)^2=br^2`

Answer

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) and the ratio of its roots. Let's denote the roots of the equation as \( \alpha \) and \( \beta \). Given that the ratio of the roots is \( r \), we can express the roots in terms of \( r \):

1. **Express the roots in terms of r**:
\[ \alpha = r\beta \]

2. **Use the sum of roots**:
The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting \( \alpha = r\beta \):
\[ r\beta + \beta = -\frac{b}{a} \]
Factoring out \( \beta \):
\[ \beta(r + 1) = -\frac{b}{a} \]
Thus, we can express \( \beta \) as:
\[ \beta = -\frac{b}{a(r + 1)} \]

3. **Use the product of roots**:
The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} \]
Substituting \( \alpha = r\beta \):
\[ r\beta \cdot \beta = \frac{c}{a} \]
This simplifies to:
\[ r\beta^2 = \frac{c}{a} \]
Therefore, we can express \( \beta^2 \) as:
\[ \beta^2 = \frac{c}{ar} \]

4. **Substituting \( \beta^2 \) into the sum of roots equation**:
Now, substituting \( \beta^2 = \frac{c}{ar} \) into the equation derived from the sum of roots:
\[ \beta^2(r + 1)^2 = \left(-\frac{b}{a}\right)^2 \]
Thus:
\[ \frac{c}{ar}(r + 1)^2 = \frac{b^2}{a^2} \]

5. **Cross-multiplying to eliminate fractions**:
Cross-multiplying gives:
\[ c(r + 1)^2 = \frac{b^2}{a} \cdot r \]
Rearranging this leads to:
\[ ac(r + 1)^2 = b^2r \]

6. **Identifying the correct option**:
From the derived equation \( ac(1 + r)^2 = rb^2 \), we can see that this matches with option B from the original question.

Thus, the correct answer is:
\[ \text{Option B: } ac(1 + r)^2 = rb^2 \]