We have been doing Conic Sections in math class (Dealing with Straightlines, Circles, Parabola, Ellipses, and Hyperbolas), and I came across a set of conditions that help to recognize when a general equation of conic $$Conic:( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0) - [1]$$ represents the different kinds of sections. My Textbook reports:
- a) the conditions for $[1]$ to represent an ellipse are $Δ≠0$ and $h^2<ab$ for $0<e<1$, and
- b) the conditions for $[1]$ to represent an empty set are also $Δ≠0$ and $h^2<ab$.
(Where $Δ = Discriminant = abc + 2fgh -af^2 -bg^2 - ch^2$, $e = eccentricity$ of the conic)
The two conditions seem to be the same, I feel there might be a missing condition here to distinguish between the two (An Ellipse and an empty set). It would be helpful if you could tell me if there is a missing condition that my textbook missed out on, or if there isn't one then if you could help me distinguish between the two. Thanks!
Write the general equation using a $2 \times 2$ matrix $Q$ and a $2 \times 1$ vector $v$ and a constant $c$. Starting with
$ a x^2 + 2 h xy +b y^2 + 2 g x + 2 f y + c = 0 $
by defining the vector $r = [x, y]^T$, this becomes
$r^T Q r + v^T r + c = 0 $
where
$Q = \begin{bmatrix} a && h \\ h && b \end{bmatrix} $
$v = \begin{bmatrix} 2 g \\ 2 f \end{bmatrix} $
The condition to have an ellipse or an empty set is that the eigenvalues of $Q$ have the same sign. The eigenvalues of $Q$ are the roots of this quadratic equation
$ (\lambda - a)(\lambda - b) - h^2 = 0 $
And they will have the same sign if $ a b - h^2 \gt 0$, i.e. $h^2 \lt a b $