# Is a circle classified as an ellipse?

I read that an ellipse had $2$ focal points. So, I thought if a circle had $2$ points that were simply infinitesimally close together wouldn't it be classified as an ellipse?

• Yes, you can say that circle is a special case of an ellipse with coinciding semiaxis. Apr 25 '15 at 6:20
• A circle is simply a degenerate ellipse, one in which the two focal points coincide. Apr 25 '15 at 6:21
• I think to say that the foci are infinitesimally close is not a good way to say it. In circle, the foci are just the same point. Apr 25 '15 at 6:26
• Not sure if this should be asked in another question but can you draw an ellipse (that is not a circle) using a a pair of compasses? Apr 25 '15 at 6:43
• @anonymous No. You can draw an ellipse using two pins and a string though.
– DRF
Apr 25 '15 at 8:19

Yes. A circle is a special case of an ellipse.

The equation of an ellipse centered at the origin $(0,0)$ is: $$\left(\frac xa \right)^2 + \left(\frac yb \right)^2=1$$

When $a=b=1$, this gives the equation of the unit circle: $x^2+y^2=1$.

In general, if $a=b=r$, you get the equation of a circle with radius $r$: $$x^2+y^2=r^2$$

Can you come up the equation of a circle not centered at $(0,0)$, but instead centered at $(h,k)$?

Both the foci of a circle coincide and thus, its eccentricity is zero. So yes, it is an ellipse.
It is like that all squares are rectangles but all rectangles are not squares.