If r is the ratio of the roots of `ax ^2 + bx +C=0` then `((r +1)^2)/(r )`=

Options

A. `(b)/(ac)`
B. `(2b)/(a)`
C. `(b^(2))/(ac)`
D. `(2b)/(c)`

Answer

To solve the problem, we need to find the expression \(\frac{(r + 1)^2}{r}\) where \(r\) is the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\).

### Step-by-Step Solution

1. **Understanding the Roots**:
For the quadratic equation \(ax^2 + bx + c = 0\), let the roots be \(\alpha\) and \(\beta\). According to Vieta's formulas:
\[ \alpha + \beta = -\frac{b}{a} \quad \text{(1)} \]
\[ \alpha \beta = \frac{c}{a} \quad \text{(2)} \]

2. **Expressing Roots in Terms of Ratio**:
Given that \(r\) is the ratio of the roots, we can express the roots as:
\[ \alpha = r\beta \quad \text{(3)} \]

3. **Substituting into the Product of Roots**:
Substitute equation (3) into equation (2):
\[ (r\beta) \beta = \frac{c}{a} \]
This simplifies to:
\[ r\beta^2 = \frac{c}{a} \]
Rearranging gives:
\[ \beta^2 = \frac{c}{ar} \quad \text{(4)} \]

4. **Substituting into the Sum of Roots**:
Now substitute equation (3) into equation (1):
\[ r\beta + \beta = -\frac{b}{a} \]
This simplifies to:
\[ (r + 1)\beta = -\frac{b}{a} \]
Rearranging gives:
\[ \beta = -\frac{b}{a(r + 1)} \quad \text{(5)} \]

5. **Finding \(\alpha\)**:
Substitute equation (5) back into equation (3) to find \(\alpha\):
\[ \alpha = r\left(-\frac{b}{a(r + 1)}\right) = -\frac{rb}{a(r + 1)} \quad \text{(6)} \]

6. **Finding \(\alpha^2\) and \(\beta^2\)**:
Now, we can find \(\alpha^2\) and \(\beta^2\):
From (4):
\[ \beta^2 = \frac{c}{ar} \]
From (6):
\[ \alpha^2 = \left(-\frac{rb}{a(r + 1)}\right)^2 = \frac{r^2b^2}{a^2(r + 1)^2} \]

7. **Using the Sum of Squares**:
We know:
\[ \alpha^2 + \beta^2 + 2\alpha\beta = \left(-\frac{b}{a}\right)^2 \]
Substituting the values:
\[ \frac{r^2b^2}{a^2(r + 1)^2} + \frac{c}{ar} + 2\left(-\frac{rb}{a(r + 1)}\right)\left(-\frac{b}{a}\right) = \frac{b^2}{a^2} \]

8. **Simplifying**:
After simplifying, we can derive:
\[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

### Conclusion
Thus, the final result is:
\[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]