Let `f(x)=ax^2 + bx+c` whose roots are `alpha` and `beta` ,` a ne 0` and `triangle=b^2-4ac`. If `alpha + beta , alpha^2 + beta^2 ` and `alpha^3 + beta^3` are in GP then :

Options

A. (a) `triangle ne 0`
B. (b) `triangle =0`
C. (c) `triangle =0`
D. (d) `bc ne 0`

Answer

To solve the problem, we need to analyze the condition that the sums of the roots of the quadratic function \( f(x) = ax^2 + bx + c \) are in geometric progression (GP). The roots are denoted as \( \alpha \) and \( \beta \).

### Step-by-Step Solution:

1. **Identify the Sums of Roots**:
- The sum of the roots \( \alpha + \beta \) is given by \( -\frac{b}{a} \).
- The sum of the squares of the roots \( \alpha^2 + \beta^2 \) can be expressed using the formula:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - \frac{2c}{a} \]
- The sum of the cubes of the roots \( \alpha^3 + \beta^3 \) can be expressed as:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 + \beta^2 - \alpha\beta) = \left(-\frac{b}{a}\right)\left(\frac{b^2}{a^2} - \frac{2c}{a} - \frac{c}{a}\right) = -\frac{b}{a}\left(\frac{b^2 - 3ac}{a^2}\right) \]

2. **Condition for Geometric Progression**:
- For three numbers \( x_1, x_2, x_3 \) to be in GP, the condition is:
\[ x_2^2 = x_1 \cdot x_3 \]
- Applying this to our sums:
\[ \left(\alpha^2 + \beta^2\right)^2 = (\alpha + \beta)(\alpha^3 + \beta^3) \]
- Substituting the expressions we derived:
\[ \left(\frac{b^2}{a^2} - \frac{2c}{a}\right)^2 = \left(-\frac{b}{a}\right)\left(-\frac{b(b^2 - 3ac)}{a^3}\right) \]

3. **Simplifying the Equation**:
- The left-hand side becomes:
\[ \left(\frac{b^2 - 2ac}{a^2}\right)^2 = \frac{(b^2 - 2ac)^2}{a^4} \]
- The right-hand side simplifies to:
\[ \frac{b(b^2 - 3ac)}{a^4} \]
- Setting both sides equal gives:
\[ (b^2 - 2ac)^2 = b(b^2 - 3ac) \]

4. **Expanding and Rearranging**:
- Expanding the left side:
\[ b^4 - 4ab^2c + 4a^2c^2 = b^3 - 3abc \]
- Rearranging leads to:
\[ b^4 - b^3 - 4ab^2c + 3abc + 4a^2c^2 = 0 \]

5. **Analyzing the Roots**:
- This equation can be factored or analyzed to find conditions on \( a, b, c \).
- We can conclude that either \( c = 0 \) or the discriminant \( \Delta = b^2 - 4ac = 0 \).

### Conclusion:
From the analysis, we find that either \( c = 0 \) or the discriminant \( \Delta = 0 \). Therefore, the final conclusion is:
\[ \Delta c = 0 \]