If `alpha, beta` are the roots of `ax^(2) +bx+c=0 and alpha + beta, alpha^(2)+beta^(2) , alpha^(3)+beta^(3)` are in G.P., then

Options

A. `Delta ne 0`
B. `b Delta =0`
C. `cb ne 0`
D. `c Delta =0`

Answer

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \). We are given that \( \alpha + \beta \), \( \alpha^2 + \beta^2 \), and \( \alpha^3 + \beta^3 \) are in geometric progression (G.P.).

### Step 1: Identify the relationships between the roots
From Vieta's formulas, we know:
- The sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} \]
- The product of the roots:
\[ \alpha \beta = \frac{c}{a} \]

### Step 2: Express \( \alpha^2 + \beta^2 \) and \( \alpha^3 + \beta^3 \)
We can express \( \alpha^2 + \beta^2 \) using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values from Vieta's formulas:
\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - \frac{2c}{a} = \frac{b^2 - 2ac}{a^2} \]

Next, we can express \( \alpha^3 + \beta^3 \) using the identity:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 + \beta^2 - \alpha\beta) \]
Substituting the known values:
\[ \alpha^3 + \beta^3 = \left(-\frac{b}{a}\right)\left(\frac{b^2 - 2ac}{a^2} - \frac{c}{a}\right) \]
This simplifies to:
\[ \alpha^3 + \beta^3 = -\frac{b}{a} \left(\frac{b^2 - 2ac - ac}{a^2}\right) = -\frac{b(b^2 - 3ac)}{a^3} \]

### Step 3: Setting up the G.P. condition
Since \( \alpha + \beta \), \( \alpha^2 + \beta^2 \), and \( \alpha^3 + \beta^3 \) are in G.P., we have the condition:
\[ (\alpha^2 + \beta^2)^2 = (\alpha + \beta)(\alpha^3 + \beta^3) \]
Substituting the expressions we derived:
\[ \left(\frac{b^2 - 2ac}{a^2}\right)^2 = \left(-\frac{b}{a}\right)\left(-\frac{b(b^2 - 3ac)}{a^3}\right) \]
This simplifies to:
\[ \frac{(b^2 - 2ac)^2}{a^4} = \frac{b^2(b^2 - 3ac)}{a^4} \]
Multiplying through by \( a^4 \) (assuming \( a \neq 0 \)):
\[ (b^2 - 2ac)^2 = b^2(b^2 - 3ac) \]

### Step 4: Expanding and rearranging
Expanding both sides:
\[ b^4 - 4ab^2c + 4a^2c^2 = b^4 - 3ab^2c \]
Rearranging gives:
\[ 4a^2c^2 - ab^2c + 3ab^2c = 0 \]
This simplifies to:
\[ 4a^2c^2 - ab^2c = 0 \]
Factoring out \( c \):
\[ c(4ac - b^2) = 0 \]

### Step 5: Conclusion
From this equation, we have two cases:
1. \( c = 0 \)
2. \( 4ac - b^2 = 0 \) which implies \( b^2 = 4ac \)

Thus, the required condition is:
\[ c = 0 \quad \text{or} \quad b^2 - 4ac = 0 \]