Question

If x2+y2+1/x2+1/y2=4, then the value of x2+y2 is

Answer 1

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Answer 2

Answer:

$The\: value \: of \:x^{2}+y^{2}=2$

Step-by-step explanation:

$Given\: x^{2}+y^{2}+\frac{1}{x^{2}}+\frac{1}{y^{2}}=4$

$\implies x^{2}+\frac{1}{x^{2}}+y^{2}+\frac{1}{y^{2}}-4=0$

$\implies x^{2}+\frac{1}{x^{2}}-2+y^{2}+\frac{1}{y^{2}}-2=0$

$\implies\left( x^{2}+\frac{1}{x^{2}}-2\times x \times \frac{1}{x}\right)+\left(y^{2}+\frac{1}{y^{2}}-2\times y \times \frac{1}{y}\right)=0$

$\implies \left(x-\frac{1}{x}\right)^{2}+\left(y-\frac{1}{y}\right)^{2} = 0$

$\implies \left(x-\frac{1}{x}\right)^{2}=0\: and \: \left(y-\frac{1}{y}\right)^{2} = 0$

$\implies x-\frac{1}{x}=0\:and \: y-\frac{1}{y}=0$

$\implies x =\frac{1}{x}\:and\:y=\frac{1}{y}$

$\implies x^{2}=1 \:and \: y^{2}=1$

$The\: value \: of \:x^{2}+y^{2}\\=1+1\\=2$

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