If the ratio of the roots of the equation `ax^2+bx+c=0` is equal to ratio of roots of the equation `x^2+x+1=0` then a,b,c are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Explanation

To solve the problem step by step, we need to analyze the given equations and derive the relationship between the coefficients \( a \), \( b \), and \( c \).

### Step 1: Identify the roots of the equations
Let the roots of the equation \( ax^2 + bx + c = 0 \) be \( \alpha \) and \( \beta \). According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

For the equation \( x^2 + x + 1 = 0 \), let the roots be \( \gamma \) and \( \delta \):
- The sum of the roots \( \gamma + \delta = -1 \)
- The product of the roots \( \gamma \delta = 1 \)

### Step 2: Set up the ratio of the roots
We are given that the ratio of the roots of the first equation is equal to the ratio of the roots of the second equation:
\[ \frac{\alpha}{\beta} = \frac{\gamma}{\delta} \]
This implies:
\[ \alpha \delta = \beta \gamma \]

### Step 3: Express the ratios in terms of sums and products
From the ratios, we can also express:
\[ \frac{\alpha}{\beta} + 1 = \frac{\gamma}{\delta} + 1 \]
This leads to:
\[ \frac{\alpha + \beta}{\beta} = \frac{\gamma + \delta}{\delta} \]

### Step 4: Substitute the values from Vieta's formulas
Substituting the values from Vieta's:
\[ \frac{-\frac{b}{a}}{\beta} = \frac{-1}{\delta} \]
and
\[ \frac{\beta}{\alpha} + 1 = \frac{\delta}{\gamma} + 1 \]
This leads to:
\[ \frac{\beta + \gamma}{\alpha} = \frac{\delta + \gamma}{\gamma} \]

### Step 5: Add the two equations
Adding the two derived equations:
\[ \frac{\alpha + \beta}{\beta} + \frac{\beta + \gamma}{\alpha} = \frac{\gamma + \delta}{\delta} + \frac{\delta + \gamma}{\gamma} \]

### Step 6: Simplify the equation
This can be simplified to:
\[ \frac{\alpha + \beta}{\alpha \beta} = \frac{\gamma + \delta}{\gamma \delta} \]

### Step 7: Substitute the known sums and products
Substituting the known values:
\[ \frac{-\frac{b}{a}}{\frac{c}{a}} = \frac{-1}{1} \]
This simplifies to:
\[ \frac{b^2}{ac} = 1 \]

### Step 8: Final relationship
From the above equation, we conclude:
\[ b^2 = ac \]
Thus, we can say that \( a, b, c \) are in geometric progression.