Question

If $x=9+4\sqrt{5}$ and xy=1 then find $\frac{1}{x^2}+\frac{1}{y^2}$

Answer 1

Given $x=9+4\sqrt{5}$

and also $xy=1⟹y=\frac{1}{x}=\frac{1}{9+4\sqrt{5}}=\frac{9-4\sqrt{5}}{(9{)}^2-(4\sqrt{5}{)}^2}=\frac{9-4\sqrt{5}}{81-80}=9-4\sqrt{5}$

$x+y=9+4\sqrt{5}+9-4\sqrt{5}=18$

Squaring on both sides

$⟹(x+1{)}^2=324$

$⟹x^2+y^2+2xy=324$

$⟹x^2+y^2=322$

$\frac{1}{x^2}+\frac{1}{y^2}=\frac{x^2+y^2}{x^2{y}^2}=\frac{322}{(1{)}^2}=322$