Question

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

• A. 81
• B. 322
• C. 97
• D. 2

Answer 1

Answer: (B)

322

Given $x=9+4\sqrt{5}andxy=1$
$y=\frac{1}{x}$
$\therefore y=\frac{1}{9+4\sqrt{5}}$
By rationalization
$y=\frac{1}{9+4\sqrt{5}}\times \frac{9-4\sqrt{5}}{9-4\sqrt{5}}$
$y=\frac{9-4\sqrt{5}}{81-80}=9-4\sqrt{5}$
Then
$=\frac{1}{x^2}+\frac{1}{y^2}$
$=\frac{x^2+y^2}{{\left(xy\right)}^2}$
$=\frac{{(9+4\sqrt{5})}^2+{(9-4\sqrt{5})}^2}{{\left(1\right)}^2}$
$\Rightarrow \frac{81+72\sqrt{5}+80+81-72\sqrt{5}+80}{1}$
$\Rightarrow 322$