Ellipse equation

Question

Having the equation:

$ax^2+by^2+cxy=d$,

where $a,b,c$ and $d$ are a constants. In which case (or which condition on $a,b,c$ and/or $d$) the above equation describes an ellipse ?

Answer 1

It is an ellipse in the case $c^2-4ab<0$ and $d$ has the same sign as $a$ or $b$. The signs of $a$ and $b$ are the same for $c^2-4ab<0$. The condition on $d$ is necessary to get an actual ellipse in the real domain.

Answer 2

Proof of the condition given by @Alessandro Blasetti

Let us assume that $a>0$ WLOG (we can multiply both sides by $-1$).

The canonical form:

$$ax^2+by^2+cxy=a\left(x+\frac{c}{2a}y\right)^2+k y^2=d \ \ \ \text{with} \ \ k=\dfrac{4ab-c^2}{4a}$$

shows that, up to an affine transform (which does no change the nature of the conic section), we recognize the form $AX^2+BY^2=d$ of an ellipse if $k>0$, i.e., $4ab-c^2 >0$ (a contrario, the form $AX^2-BY^2=d$ of an hyperbola).