A Lemniscate Checklist

The curve $C$ (a lemniscate) is the locus of points whose distances to $F_1(-a, 0)$ and $F_2(a, 0)$ have product $a^2$. Determine, with proof, which of the following are true:

1. if $a = 1$, the point $(\sqrt{2}, 0)$ lies on $C$; 2. if $a = 1$, the equation of $C$ is $\left(x^2 + y^2\right)^2 = 2x^2 - y^2$; 3. if $a = 2$, the maximum $y$-coordinate of points on $C$ is $1$; 4. if $(x_0, y_0)$ lies on $C$, then $|y_0| \leqslant |x_0|$.