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Your curves are the plots of $\log(x\sqrt{y})=0$ and $\log(x^2y^2)=0$ respectively. The equation $\log(x\sqrt{y})=\log(x^2y^2)$ is more general, and has an infinity of solutions:

sage: (log(x*sqrt(y))==log(x^2*y^2)).log_expand().solve(y)
[y == x^(-2/3)]

The intersection point of your curves is a special case of this general solution, where both left-hand and right-hand of the equation are null ; this is a system of two equations, which Sage can solve:

sage: solve([log(x*sqrt(y))==0,log(x^2*y^2)==0],[x,y])
[[x == 1, y == 1]]

HTH,