Are every two special ellipses congruent to each other?

Question

A special ellipse is an ellipse whose area is equal to its perimeter.

Are there two special ellipses which are not congruent to each other? Are there infinite number of special ellipses which are mutually non congruent?

Answer 1

That can be answered by a scaling argument.

For any $a>1$, consider the ellipse $E_a =\{ \frac{x^2}{a^2} + a^2y^2 =1\}$. Then for all $r>0$,

$$\operatorname{Area} (rE_a) = r^2 \operatorname{Area} (E_a) = \pi r^2 , \operatorname{Perimeter} (rE_a) = r \operatorname{Perimeter} (E_a)$$

Thus choosing $r_a = \operatorname{Perimeter} (E_a)/\pi$ give a special ellipse $r_a E_a$. For different $a_1 \neq a_2$, $r_{a_1} E_{a_1}$ is not congruent to $r_{a_2} E_{a_2}$.