Finding conditions for a point residing in the interior of an ellipse

Question

I have an ellipse with the equation $x^2+2y^2-2xy-1=0$. Suppose $(h,k)$ is a point residing in the interior region of the ellipse. Should this point satisfy any condition in terms of $h,k$?

Answer 1

In general, if $\mathcal{E}$ is an ellipsoid in $\mathbb{R}^n$, defined by $$ \mathcal{E}\colon\mathbf{x}^T\Sigma\mathbf{x}=1, $$ where $\mathbf{x}\in\mathbb{R}^n$ and $\Sigma\in\mathbb{S}_{++}^n$, i.e., $\Sigma$ is a positive definite matrix, then, a point $\mathbf{k}\in\mathbb{R}^n$ resides in the interior region of the ellipsoid, iff $$ \mathbf{k}^T\Sigma\mathbf{k}<1. $$