Question

If x=9+4√5 and xy=1 ,then find 1/x ki power 2 + 1/y ki power 2

Answer 1

$\frac{1}{ {x}^{2} } + \frac{1}{ {y}^{2} } = ( { \frac{1}{x} + \frac{1}{y} )}^{2} - \frac{2}{xy} \\ = ( \frac{1}{9 + 4 \sqrt{5} } + (9 + 4 \sqrt{5}) )power \: 2 - 2 \\ = {( \frac{1 + {(9 + 4 \sqrt{5} )}^{2} }{9 + 4 \sqrt{5} }) }^{2} - 2 \\ = { (\frac{1 + 81 +80 + 72 \sqrt{5} }{9 + 4 \sqrt{5} }) }^{2} - 2 \\ = (\frac{ {162 + 72 \sqrt{5} \times (9 - 4 \sqrt{5)} }^{2} }{(9 + 4 \sqrt{5})(9 - 4 \sqrt{5} )} ) - 2 \\ = {( \frac{1458 + 648 \sqrt{5} - 648 \sqrt{5} - 1440}{81 - 80} )}^{2} - 2 \\ = {18}^{2} - 2 \\ = 324 - 2 \\ = 322 \: \: \: \: \: ans$