The sum of the roots of the equation `x^(2)+bx+c=0` (wheere b and c are non-zero) is equal to the sum of the reciprocals of their squares. Then `(1)/(c),b,(c)/(b)` are in

Options

A. AP
B. GP
C. HP
D. None of the above

Explanation

To solve the problem, we start with the quadratic equation given:

\[ x^2 + bx + c = 0 \]

Let the roots of the equation be \( \alpha \) and \( \beta \). According to Vieta's formulas, we know:

1. The sum of the roots \( \alpha + \beta = -b \)
2. The product of the roots \( \alpha \beta = c \)

We are given that the sum of the roots is equal to the sum of the reciprocals of their squares:

\[ \alpha + \beta = \frac{1}{\alpha^2} + \frac{1}{\beta^2} \]

We can rewrite the right-hand side using the identity for the sum of squares:

\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\beta^2 + \alpha^2}{\alpha^2 \beta^2} \]

Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \), we substitute:

\[ \alpha^2 + \beta^2 = (-b)^2 - 2c = b^2 - 2c \]

Thus, we have:

\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{b^2 - 2c}{c^2} \]

Now, equating the two expressions for the sum of the roots:

\[ -b = \frac{b^2 - 2c}{c^2} \]

Cross-multiplying gives:

\[ -bc^2 = b^2 - 2c \]

Rearranging this equation leads to:

\[ b^2 + bc^2 - 2c = 0 \]

This is a quadratic equation in terms of \( b \). We can apply the quadratic formula:

\[ b = \frac{-c^2 \pm \sqrt{(c^2)^2 + 8c}}{2} \]

Next, we need to analyze the relationship between \( \frac{1}{c}, b, \frac{c}{b} \).

1. From the quadratic equation, we can find the discriminant to ensure \( b \) has real values.
2. We will check if \( \frac{1}{c}, b, \frac{c}{b} \) are in arithmetic progression (AP), geometric progression (GP), or harmonic progression (HP).

To check if \( \frac{1}{c}, b, \frac{c}{b} \) are in GP, we need to verify:

\[ b^2 = \frac{1}{c} \cdot \frac{c}{b} \]

This simplifies to:

\[ b^2 = \frac{1}{b} \]

Thus, multiplying both sides by \( b \) (assuming \( b \neq 0 \)) gives:

\[ b^3 = 1 \implies b = 1 \]

Now substituting \( b = 1 \) back into the original quadratic equation gives us the relationships needed to check the progressions.

### Conclusion:
The values \( \frac{1}{c}, b, \frac{c}{b} \) are in geometric progression (GP) if \( b^3 = 1 \).