On the positivity of the distance between two disjoint ellipsoids

Question

Let closed convex sets $P_1$ and $P_2$ be defined as follows

$$P_i := \{ x \in \Bbb R^n : x^\top A_i x + b_i^\top x + c_i \leq 0 \}$$

where $A_1$ and $A_2$ are positive semidefinite matrices. Assume $P_1 \cap P_2 = \emptyset$. Prove (or provide a counter-example) that $$d(P_1 , P_2):= \inf \{ \|x-y\| \; | \; x \in P_1 ,~ y \in P_2 \} > 0$$