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Question Number 60445 by ANTARES VY last updated on 21/May/19

Commented by bhanukumarb2@gmail.com last updated on 21/May/19

$$cauchyinequality$$

Answered by MJS last updated on 21/May/19

$$x=\sinh u$$$$y=\sinh v$$$${\mathrm{e}}^{u+v}=2019$$$$v=\ln 2019-u$$$$\sinh u=\frac{1}{2}({\mathrm{e}}^u-{\mathrm{e}}^{-u})$$$$\sinh v=\frac{1}{2}({2019\mathrm{e}}^{-u}-\frac{1}{2019}{\mathrm{e}}^u)=\frac{2019}{{2\mathrm{e}}^u}-\frac{{\mathrm{e}}^u}{4038}$$$$(x+2019y)(y+2019x)=$$$$=\frac{2038180}{{\mathrm{e}}^u}\times \frac{{2038180\mathrm{e}}^u}{2019}=\frac{4154177712400}{2019}$$

Answered by MJS last updated on 21/May/19

$$y=x$$$${(x+\sqrt{x^2+1})}^2=2019$$$$\Rightarrow x=\frac{1009}{2019}\sqrt{2019}$$$${\left(2020x\right)}^2=\frac{4154177712400}{2019}$$

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