Question

P4+q4+1/p4+1/q4=4 to √p2+q2=?

Answer 1

$Given \: p^{4} + q^{4} + \frac{1}{p^{4}} + \frac{1}{q^{4}} = 4$

$\implies p^{4} + q^{4} + \frac{1}{p^{4}} + \frac{1}{q^{4}} -4= 0$

$\implies p^{4} + \frac{1}{p^{4}}-2 + q^{4} + \frac{1}{q^{4}} -2= 0$

$(p^{2})^{2} +\Big( \frac{1}{p^{2}}\Big) ^{2} -2\times p^{2} \times \frac{1}{p^{2}} + (q^{2})^{2} +\Big( \frac{1}{q^{2}}\Big) ^{2} -2\times q^{2} \times \frac{1}{q^{2}}= 0$

$\implies \Big ( p^{2} - \frac{1}{p^{2}}\Big)^{2} + \Big ( q^{2} - \frac{1}{q^{2}}\Big)^{2} = 0$

$\implies p^{2} - \frac{1}{p^{2}} =0 \: Or \: q^{2} - \frac{1}{q^{2}} =0$

$\implies p^{2} = \frac{1}{p^{2}} \: Or \: q^{2} = \frac{1}{q^{2}}$

$\implies p^{4} = 1 \: Or \: q^{4} = 1$

$\implies p^{2} = 1 \: Or \: q^{2} = 1$

Therefore.,

$\red { Value \: of \: \sqrt{ p^{2} + q^{2} } }$

$= \sqrt{ 1 + 1 }$

$\green { = \sqrt{2} }$

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