If a,b,c are in G.P. then the roots of the equation `ax^(2)+bx+c=0` are in the ratio

Options

A. `(1)/(2) (-1+ i sqrt(3))`
B. `(1)/(2) (-1-i sqrt(3))`
C. `(1)/(2)(1+ i sqrt(3))`
D. `(1)/(2)(1-i sqrt(3))`

Answer

To solve the problem, we need to determine the ratio of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) given that \( a, b, c \) are in geometric progression (G.P.).

### Step-by-Step Solution:

1. **Understanding the Condition of G.P.**:
Since \( a, b, c \) are in G.P., we have the relationship:
\[ b^2 = ac \]

2. **Using Vieta's Formulas**:
For the quadratic equation \( ax^2 + bx + c = 0 \), by Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

3. **Expressing the Roots**:
We can express the roots in terms of a single variable. Let:
\[ \frac{\alpha}{\beta} = k \quad \text{(where \( k \) is the ratio of the roots)} \]
Thus, we can write:
\[ \alpha = k\beta \]

4. **Substituting into Vieta's Formulas**:
- From the sum of the roots:
\[ k\beta + \beta = -\frac{b}{a} \implies \beta(k + 1) = -\frac{b}{a} \implies \beta = -\frac{b}{a(k + 1)} \]
- From the product of the roots:
\[ k\beta \cdot \beta = \frac{c}{a} \implies k\beta^2 = \frac{c}{a} \]

5. **Substituting for \(\beta\)**:
Substitute \(\beta\) from the sum of roots into the product of roots:
\[ k\left(-\frac{b}{a(k + 1)}\right)^2 = \frac{c}{a} \]
Simplifying this gives:
\[ k\frac{b^2}{a^2(k + 1)^2} = \frac{c}{a} \]
Multiplying both sides by \( a^2(k + 1)^2 \):
\[ kb^2 = ca(k + 1)^2 \]

6. **Using the G.P. Condition**:
Substitute \( c = \frac{b^2}{a} \) (from \( b^2 = ac \)):
\[ kb^2 = \frac{b^2}{a} a(k + 1)^2 \implies kb^2 = b^2(k + 1)^2 \]
Dividing both sides by \( b^2 \) (assuming \( b \neq 0 \)):
\[ k = (k + 1)^2 \]

7. **Solving the Quadratic Equation**:
Rearranging gives:
\[ k^2 + 2k + 1 - k = 0 \implies k^2 + k + 1 = 0 \]
The roots of this equation can be found using the quadratic formula:
\[ k = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2} \]

8. **Conclusion**:
The ratio of the roots \( \alpha : \beta \) is given by:
\[ \frac{\alpha}{\beta} = k \quad \text{where } k = \frac{-1 \pm i\sqrt{3}}{2} \]