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Ellipses can be elegantly described in four ways.
The simplest description of an ellipse is as a squashed or stretched circle. Start with the unit circle $x^2 + y^2 =1$, and stretch it by a factor of $a$ in the $x$ direction and $b$ in the $y$ direction to get:
This fact gives elliptical rooms amazing acoustic properties. If you whisper at one focus of such a room, the sound waves from your voice will bounce off the walls and converge at the other focus  that's why it is called a focus. The same goes for light reflecting off elliptical mirrors. To understand the amazing fact, let's convert the equation $L_1 + L_2 = 2a$ to rectangular coordinates: \begin{eqnarray*} L_1 + L_2 & = & 2a \cr\cr L_1 & = & 2aL_2 \cr \cr \sqrt{(x+c)^2+y^2} & = & 2a \sqrt{(xc)^2 + y^2} \cr\cr (x+c)^2 + y^2 & = & 4a^2 + (xc)^2 + y^2  4a \sqrt{(xc)^2 + y^2}\cr\cr 4a\sqrt{(xc)^2 + y^2}&=& 4a^24cx \cr \cr a \sqrt{(xc)^2 + y^2} &=& a^2cx \cr \cr a^2(xc)^2+ a^2 y^2 &=& a^4+c^2x^2 2a^2cx \cr \cr a^2x^2 + a^2c^2 2a^2cx + y^2 &=& a^4 + c^2x^2 2a^2cx \cr \cr (a^2c^2)x^2 + a^2 y^2 &=& a^2(a^2c^2) \cr \cr b^2 x^2 + a^2 y^2 &=& a^2b^2 \cr \cr \frac{x^2}{a^2} + \frac{y^2}{b^2} &=& 1,\end{eqnarray*} where we have used the fact that $b^2=a^2c^2$. That's a long and messy calculation for a simple and elegant result. You should be able to construct the equation of an ellipse given any two of $a$, $b$ and $c$, since you can get the third from $c^2=a^2b^2.$
