The circle as a degenerate ellipse

What happens to the ellipse when the two foci move together until they coincide? The locus does not disappear: it turns into a figure we already know.

Observation — The circle as a limiting case

If the two foci $F_1$ and $F_2$ coincide (that is $c = 0$), the definition of the ellipse becomes $d(P,F) + d(P,F) = 2a$, that is $d(P,F) = a$: it is a circle with centre $F$ and radius $a$. The canonical form becomes again $\tfrac{x^2}{a^2}+\tfrac{y^2}{a^2}=1$, that is $x^2+y^2 = a^2$. The parameter $b$ becomes equal to $a$, and the eccentricity $e = c/a = 0$. Hence the circle is a limiting case of the ellipse, the one of maximum “roundness”.

Links

Topics: Ellipse
Concepts: Circle · Eccentricity · Ellipse · Foci