Question-197064

Question Number 197064 by Amidip last updated on 07/Sep/23

Answered by witcher3 last updated on 07/Sep/23

x \sqrt{1 + {(\frac{y}{x})}^{\frac{2}{3}}} + y {(1 + {(\frac{y}{x})}^{\frac{2}{3}})}^{\frac{1}{2}} = a

a^{2} = (x^{2} + y^{2} + x^{2} {(\frac{y}{x})}^{\frac{2}{3}} + y^{2} {(\frac{x}{y})}^{\frac{2}{3}})

+ 2 \sqrt{{2 x}^{2} y^{2} + x^{2} y^{2} ({(\frac{x}{y})}^{\frac{2}{3}} + {(\frac{y}{x})}^{\frac{2}{3}})}

= x^{2} + y^{2} + 2 xy ({(\frac{x}{y})}^{\frac{1}{3}} + {(\frac{y}{x})}^{\frac{1}{3}}) + x^{\frac{4}{3}} y^{\frac{2}{3}} + y^{\frac{4}{3}} x^{\frac{2}{3}}

x^{2} + y^{2} + {3 x}^{\frac{2}{3}} y^{\frac{4}{3}} + {3 y}^{\frac{2}{3}} x^{\frac{4}{3}}

= {(x^{\frac{2}{3}} + y^{\frac{2}{3}})}^{3} = a^{2} \Rightarrow x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}

Answered by som(math1967) last updated on 07/Sep/23

\sqrt{x^{\frac{4}{3}} (x^{\frac{2}{3}} + y^{\frac{2}{3}})} + \sqrt{y^{\frac{4}{3}} (y^{\frac{2}{3}} + x^{\frac{2}{3}})} = a

\Rightarrow x^{\frac{2}{3}} {(x^{\frac{2}{3}} + y^{\frac{2}{3}})}^{\frac{1}{2}} + y^{\frac{2}{3}} {(x^{\frac{2}{3}} + y^{\frac{2}{3}})}^{\frac{1}{2}} = a

\Rightarrow (x^{\frac{2}{3}} + y^{\frac{2}{3}}) {(x^{\frac{2}{3}} + y^{\frac{2}{3}})}^{\frac{1}{2}} = a

\Rightarrow {(x^{\frac{2}{3}} + y^{\frac{2}{3}})}^{\frac{3}{2}} = a

∴ (x^{\frac{2}{3}} + y^{\frac{2}{3}}) = a^{\frac{2}{3}}