Has anyone come across this representation of an ellipse: $\mathcal E := \left\{ x \mid f(x) \leq 0 \right\}$, where
$$ \begin{aligned} f(x) &= x^{\top} c x+2 d^{\top} x + e\\ &=\left[\begin{array}{ll}x \\ 1\end{array}\right]^{\top}\left[\begin{array}{ll}c & d \\ d^{\top} & e\end{array}\right]\left[\begin{array}{c}x \\ 1\end{array}\right]\end{aligned} $$
I have no idea how to derive this from the conventional form:
$$\mathcal E = \left\{x \mid\left(x-x_{c}\right)^{\top} A^{-1}\left(x-x_{c}\right) \leq 1\right\}$$
Any help will be appreciated.