$\mu\in\mathbb{R},\sigma^2\gt0$
$X$ and $Y$ are random variables which are independent each other and follow $N(\mu,\sigma^2)$ and $N(0,\sigma^2)$.
To test $$H_0:\mu=0 \\\ H_1:\mu\neq0$$
we consider the rejection area $$R=\{(x,y) \in \mathbb{R}^2|x^2\geq cy^2\}$$
that is , when $(X,Y) \in R$, reject $H_0$.
$c$ is a constant value.
When we set a level of significance $\alpha(0\lt\alpha\lt1)$, what is a value of $c$?
I set $$P(X^2\geq cY^2|\mu=0)=\alpha$$ Then if I suppose $H_0$ is correct, $X$ and $Y$ has the same distribution,
then $$P(Y^2\geq cY^2) =\alpha \\\ P(1\geq c)=\alpha$$
I got stuck here, but could not see where I went wrong.