Why do the foci of an ellipse need to be on the same axis?

Question

My understanding of a definition of an ellipse is that it is the set of all points such that the sum of the Euclidean distance from the point to two foci $f_1$ and $f_2$ is equal. Or, $\{x \in \mathbb{R^2} \vert d(x,f_1) +d(x,f_2)=a\}$, for some constant $a.$

I keep seeing textbooks then say that the two foci have to be on the same (major) axis, but I don't see why this is the case. Why couldn't we have two foci on completely different axises with this definition?

For instance, look at this graph I made. Is this an ellipse, and if not then what is it?

Answer 1

The general equation of an ellipse (centered on the origin) with principal axes of length $a$ and $b$ rotated by angle $\theta$ is:

$$\frac{(x \cos \theta + y \sin \theta)^2}{a^2} + \frac{(x \sin \theta - y \cos \theta)^2}{b^2} = 1$$

If you want to displace the center, replace $x$ and $y$ by $(x - x_0)$ and $(y - y_0)$, respectively.

Note here: "axes" refers to the axes of the ellipse---NOT (necessarily) the $x$ and $y$ axes of a coordinate system. In fact, the two foci (which can be anywhere) define the major axis of the ellipse, so of course the foci must be on the same axis!

Answer 2

Another general equation is $$ Ax^2+Bxy+Cy^2+Dx+Ey+F=0\,, $$ in which the graph is an ellipse if $B^2-4AC<0$ and if the graph has at least two points (could be empty or a singleton, as you see in the cases $x^2+y^2+F=0$ when $F$ is positive or zero, respectively.