Find the value
\[ x = 7 + 4\sqrt{3}, \quad xy = 1 \]
Solution:
\[ y = \frac{1}{x} = 7 – 4\sqrt{3} \]
\[ x + y = (7 + 4\sqrt{3}) + (7 – 4\sqrt{3}) = 14 \]
\[ xy = 1 \]
\[ x^2 + y^2 = (x + y)^2 – 2xy = 14^2 – 2 = 196 – 2 = 194 \]
\[ \frac{1}{x^2} + \frac{1}{y^2} = x^2 + y^2 = 194 \]
\[ \therefore \text{Correct Answer: } 194 \ (\text{Option c}) \]