Let a,b,c be real numbers in G.P. such that a and c are positive , then the roots of the equation ` ax^(2) +bx+c=0`

Options

A. Are real and in the ratio b : ac
B. Are real
C. Are imaginary are in ration ` 1 : omega` is a non-real complex cubic root of constant
D. Are imaginary and are in the ration ` omega^(2) :1 ` with usual notation

Answer

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) under the condition that \( a, b, c \) are in geometric progression (G.P.) and \( a, c > 0 \).

### Step-by-step Solution:

1. **Understanding G.P. Condition**:
Since \( a, b, c \) are in G.P., we can express this condition mathematically as:
\[ b^2 = ac \]

2. **Given Conditions**:
We know that \( a > 0 \) and \( c > 0 \). Therefore, since both \( a \) and \( c \) are positive, it follows that \( ac > 0 \). This implies:
\[ b^2 = ac > 0 \implies b^2 > 0 \implies b \neq 0 \]

3. **Discriminant of the Quadratic Equation**:
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be determined using the discriminant \( D \):
\[ D = b^2 - 4ac \]

4. **Substituting the G.P. Condition**:
Substitute \( b^2 = ac \) into the discriminant:
\[ D = ac - 4ac = -3ac \]

5. **Analyzing the Discriminant**:
Since \( a > 0 \) and \( c > 0 \), it follows that:
\[ -3ac < 0 \]
Thus, the discriminant \( D < 0 \), which indicates that the roots of the quadratic equation are imaginary.

6. **Finding the Roots**:
The roots of the quadratic equation can be expressed as:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting \( D = -3ac \):
\[ x = \frac{-b \pm \sqrt{-3ac}}{2a} = \frac{-b \pm i\sqrt{3ac}}{2a} \]

7. **Expressing the Roots**:
Let’s denote the roots as:
\[ \alpha = \frac{-b + i\sqrt{3ac}}{2a}, \quad \beta = \frac{-b - i\sqrt{3ac}}{2a} \]

8. **Finding the Ratio of Roots**:
The ratio of the roots \( \frac{\alpha}{\beta} \) can be calculated as follows:
\[ \frac{\alpha}{\beta} = \frac{\frac{-b + i\sqrt{3ac}}{2a}}{\frac{-b - i\sqrt{3ac}}{2a}} = \frac{-b + i\sqrt{3ac}}{-b - i\sqrt{3ac}} \]
Multiplying numerator and denominator by the conjugate of the denominator:
\[ \frac{\alpha}{\beta} = \frac{(-b + i\sqrt{3ac})(-b + i\sqrt{3ac})}{(-b - i\sqrt{3ac})(-b + i\sqrt{3ac})} \]

9. **Simplifying the Expression**:
The denominator simplifies to:
\[ (-b)^2 + (i\sqrt{3ac})^2 = b^2 + 3ac \]
The numerator simplifies to:
\[ (-b + i\sqrt{3ac})^2 = b^2 - 2bi\sqrt{3ac} - 3ac \]
Thus, we have:
\[ \frac{\alpha}{\beta} = \frac{b^2 - 3ac - 2bi\sqrt{3ac}}{b^2 + 3ac} \]

10. **Conclusion**:
The ratio of the roots is a complex number, and since it can be expressed in terms of \( \omega \) (the cube roots of unity), we conclude that the roots are related to the cube roots of unity.