Given below are two statements:
Statement I: Experimentally determined oxygen-oxygen bond lengths in the \( O_2 \) are found to be the same and the bond length is greater than that of a \( O=O \) (double bond) but less than that of a single \( O-O \) bond.
Statement II: The strong lone pair-lone pair repulsion between oxygen atoms is solely responsible for the fact that the bond length in ozone is smaller than that of a double bond \( O=O \) but more than that of a single bond \( O-O \).
In light of the above statements, choose the correct answer from the options given below:
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: