Say I have a simple 1-D quadratic function $y = a_2x^2 + a_1x + a_0$. I know that if $a$ is positive, there will a unique solution for the turning point, $2$ solutions for any $y$ value above the turning point, and $0$ solutions for $y$ values below the turning point.
Now generalized to arbitrary dimension, i.e. $y = x^TAx - b^Tx + c$, where $x \in \mathbb{R}^n$, the coefficients $A \in \mathbb{R}^{n\times n}$, $b \in \mathbb{R}^n$, and $c$ is a scalar. We can assume $A$ is a symmetric matrix. I am interested in finding the solutions to $x$ given any $y$ value. If my intuitions are correct, there will be infinite solutions for $n \geq 2$ for $y$ values that are above the point where the derivative is $0$. In a 2-D case for example, the solutions will lie on the circle/ellipse formed by a plane cutting the quadratic surface at the height $y$.
How do I compute the length of this circle/ellipse for the 2-D case? Also in higher dimensions, the area (I'm not sure how to describe?) of the solution.
