The Ellipse as a Stretched Circle

Let

and

be positive numbers. To each point

in the plane we associate the point

. This association will stretch the first coordinate by

and the second coordinate by

. The applet below explores the effect of this mixed dilation on the unit circle

Let

be a point on the unit circle and let

. Then

satisfies

. On the other hand, if

satisfies the equation above, we can let

and

. This shows that

is the image of

under the mixed dilation

. Thus the image of the unit circle is the set of points

satisfying the equation of an ellipse

. Reference Lang, Serge. (1988). Basic Mathematics. Springer-Verlag.

New Resources