A Tilted Conic

Consider the curve $C \colon ax^2 + by^2 + xy = 1$, where $a, b$ are constants. Determine, with proof, which of the following are true:

1. the curve $C$ is symmetric about the origin; 2. when $a + b = 0$, the curve $C$ always passes through two fixed points; 3. there exist $a > 0$ and $b < 0$ such that the distance $|PQ|$ between two moving points $P, Q$ on $C$ has a maximum value; 4. writing $S$ for the area of the region bounded by the part of $C$ in the first quadrant and the coordinate axes: for every $a > 0$ there exists $b > 0$ such that $S > \dfrac{1}{2\sqrt{ab}}$.