If \(\log _{\left(x^{2}\right)} y+\log _{\left(y^{2}\right)} x=1\) and \(y=x^{2}-30\), then the value of \(x^{2}+y^{2}\) is

The value of $x^2+y^2$ is $72$. Let $t=\log y/\log x$, which is nonzero in the logarithm domain, so the equation becomes $t+1/t=2$ and gives $t=1$. Thus $y=x$, and $x^2-x-30=0$ gives the positive root $x=y=6$. The value $12$ adds $x$ and $y$ without squaring either.