Dear Uncle Colin

I know that $\br{\frac{x^2}{4}} + \br{\frac{y^2}{2}}=1$ is an ellipse, but I have $\br{\frac{x^2}{4}} + \br{\frac{y^2}{4}}=1$. Is that also an ellipse?

Explain Loci Like I’m a Precocious Student (Elementary)

Hi, ELLIPSE, and thanks for your message!

The short answer is yes, that is also an ellipse.

The longer answer is yes, that’s a special case of an ellipse, where the major and minor axis are equal – which makes it a circle.

Now, there are (probably) some who say “that’s not an ellipse! It’s a circle!” However, I am a strong believer in inclusive definitions (a shape can be more than one thing!)

For example, a square is both a rectangle and a rhombus – a rectangle is a quadrilateral with four right angles, and a square satisfies that property; a rhombus is a quadrilateral with four equal sides, and that fits a square too.

In the same way, an ellipse is the intersection of a cone with a plane. That’s also true of a circle, in the special case where the plane is perpendicular to the axis of the cone.

Hope that helps!

- Uncle Colin