Question

if x = 7 +4 root 3 and xy = 1 , then 1/x square + 1 /y square = ?

Answer 1

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Answer 2

Answer:

$\frac{1}{x^{2}}+\frac{1}{y^{2}}=194$

Step-by-step explanation:

$Given\: x=7+4\sqrt{3}\:--(1)$

$xy = =1\implies y =\frac{1}{x}\:--(2)$

$y = \frac{1}{7+4\sqrt{3}}$

$=\frac{7-4\sqrt{3}}{(7+4\sqrt{3})(7-4\sqrt{3})}$

$=\frac{7-4\sqrt{3}}{7^{2}-\left(4\sqrt{3}\right)^{2}}$

$=\frac{7-4\sqrt{3}}{49-48}$

$=7-4\sqrt{3}\: ---(3)$

$Now,\\\frac{1}{x^{2}}+\frac{1}{y^{2}}$

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

$=y^{2}+x^{2}\\=(7-4\sqrt{3})^{2}+(7+4\sqrt{3})^{2}\\=2[7^{2}+(4\sqrt{3})^{2}]$

/* We know the algebraic identity:

(a-b)²+(a+b)² = 2(a²+b²) */

$=2(49+48)\\=2\times 97\\=194$

Therefore,

$\frac{1}{x^{2}}+\frac{1}{y^{2}}=194$

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