Conditions for the intersection of two hyperellipsoids in $\mathbb{R}^n$ to be empty?

Question

Given two hyperellipsoids in $\mathbb{R}^n$ defined by the quadratic forms $$ (x-c_1)^\top A_1(x-c_1)=1 $$

and

$$ (x-c_2)^\top A_2(x-c_2)=1 $$ with $A_1$ and $A_2$ being positive definite. What are the necessary and sufficient conditions on $A_1, A_2, c_1$ and $c_2$ such that the intersection of the two surfaces is empty?