How to prove that this set is an ellipsoid?

Question

Assuming that matrix $Q$ is positive definite, how can we show that the following set is an ellipsoid?

$$\{ x \in \mathbb R^n \mid x^TQ^{-1}x + 2 q^T x \leq r \}$$

where $q \in \mathbb R^n$. I can show it is an ellipsoid for the case $q = 0_n$, but not for the case $q \neq 0_n$.

Answer 1

Hint: multiply the terms in the expression $$ (x+Qq)^TQ^{-1}(x+Qq), $$ compare with the given one and try to rewrite it as an ellipsoid in the variable $z=x+Qq$.

What you get is

the ellipsoid centered at $Qq$: $z^TQ^{-1}z=x^TQ^{-1}x+2q^Tx+q^TQq\le r+q^TQq=\rho$ (a constant).