# Is an ellipse a circle transformed by a simple formula?

Does any ellipse $E$ have a circle $C$ such that you can obtain $E$ by transforming $C$ by a simple formula $F$? In details , both $E$ and $C$ have the same center and the axes of $E$ are the XY axes. And F moves $(x,y)$ to $( m*x , y)$ . Where m is a real number.

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Yes, but the formula is of the specific kind you mention only after rotating the ellipse and centering it at the origin. –  ShreevatsaR Nov 7 '11 at 12:58
Sure, if you scale a circle differently in the horizontal and vertical directions, you'll certainly get an ellipse. –  Ｊ. Ｍ. Nov 7 '11 at 12:59
@J.M.: Of course, it's enough to scale (a different circle) only horizontally. :-) –  ShreevatsaR Nov 9 '11 at 3:08
I think you'll see it from $${\rm ellipse:\ \ }{x^2\over a^2}+{y^2\over b^2}=1.$$
$${\rm circle:\ \ } {x^2\over b^2}+{y^2\over b^2}=1.$$
That formula for an ellipse is only valid for axis-parallel ellipses centered at the origin. Anyway, to complete it into the form the OP wants: $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ means $\displaystyle \frac{b^2}{a^2}x^2 + y^2 = b^2$, which is the circle $x'^2 + y^2 = b^2$ with $(x',y)$ sent to $(\frac{a}{b}x', y)$. –  ShreevatsaR Nov 7 '11 at 13:22