If the sum of the roots of the quadratic equation `a x^2+b x+c=0` is equl to the sum of the squares of their reciprocals, then prove that `a/c , b/a a n d c/b` are in H.P.

Explanation

To prove that \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \) are in Harmonic Progression (H.P.), given that the sum of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is equal to the sum of the squares of their reciprocals, we can follow these steps:

### Step 1: Identify the roots and their properties
Let the roots of the quadratic 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 \( \alpha \beta = \frac{c}{a} \)

**Hint:** Recall Vieta's formulas to relate the coefficients of the polynomial to the roots.
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