Question

If $x=5+2\sqrt{6}$ and $y=5-2\sqrt{6}$ then find the value of $\frac{1}{x^2}+\frac{1}{y^2}$.

• A. $1$
• B. $100$
• C. $96$
• D. $98$

Answer 1

Answer: (D)

$98$

$x=5+2\sqrt{6}$
$=>x^2={(5+2\sqrt{6})}^2=25+24+20\sqrt{6}=49+20\sqrt{6}$
Similarly, as $y=5-2\sqrt{6}$
$=>y^2={(5-2\sqrt{6})}^2=25+24-20\sqrt{6}=49-20\sqrt{6}$
Now, $\frac{1}{x^2}+\frac{1}{y^2}=\frac{x^2+y^2}{x^2{y}^2}=\frac{49+20\sqrt{6}+49-20\sqrt{6}}{(49+20\sqrt{6})(49-20\sqrt{6})}=\frac{98}{{49}^2-{\left(20\sqrt{6}\right)}^2}=\frac{98}{1}=98$