Special Cases of an Ellipse

The standard ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ (with $b^2=a^2(1-e^2)$, $a>b$) has a natural limiting/special case worth noting.

As $b\to a$ (with $b>0$ throughout), the relation $b^2=a^2(1-e^2)$ forces $1-e^2\to1$, i.e. $e\to0$. Geometrically, as the eccentricity shrinks toward $0$, the ellipse becomes progressively more rounded — less elongated, closer to circular.

In the limiting case $a=b$ exactly, the equation $\dfrac{x^2}{a^2}+\dfrac{y^2}{a^2}=1$ becomes $x^2+y^2=a^2$ — a circle of radius $a$. At this point $e=0$, and the two foci $(\pm ae,0)$ both collapse onto a single point: the centre. …