Major and Minor Axes, Vertices and Co-Vertices of an Ellipse

Let $E$ be an ellipse with center $(0, 0)$, and let $p = (x, y)$ be a point on $E$. For which point $q$ is $|p| \le |q|$ always true?

(Note: In this case, we define $|p|$ as the distance from $p$ to the origin, or $\sqrt{x^2 + y^2}$.)