Find an equation in $x$ and $k$

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Find an equation in $x$ and $k$ if, $$6u-8v+2=k^2$$ $$u^{2}=1+2v^{2}$$ $$v=2xy$$ $$u=x^2+2xy-y^2$$ Since we have 4 equations, we can eliminate 3 variables. But somehow, I'm not able to find an equation in $x$ and $k$. Any help will be appreciated. Thanks!