\[x^{2} - 9x + 11 = 0\ \]
\[x_{1} + x_{2} = 9\]
\[x_{1} \cdot x_{2} = 11\]
\[\frac{1}{x_{1}^{2}} + \frac{1}{x_{2}^{2}} = \frac{x_{1}^{2} + x_{2}^{2}}{x_{1}^{2}x_{2}^{2}} =\]
\[= \frac{\left( x_{1} + x_{2} \right)^{2} - 2x_{1} + x_{2}}{\left( x_{1}x_{2} \right)^{2}} =\]
\[= \frac{9^{2} - 2 \cdot 11}{11^{2}} =\]
\[= \frac{81 - 22}{121} = \frac{59}{121}\]