Conic Sections

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====== Conic Sections ======

General equation for a conic equation: $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$.

This can be used to describe:
  * ellipse 
  * hyperbola
  * parabola
  * degenerate cases
    * point
    * lines
    * parallel lines
    * crossing lines

We're focusing on __non-rotated conics__. ($B=0$)

===== Parabola =====

  * 1 focal point 
  * directrix

A parabola is the locus of points such that $d_1$, the distance from the focus, and $d_2$, the distance from the directrix, are equal.

$$d_1 = d_2$$

$$\begin{align} y+c & = \sqrt{x^{2}+(y-c)^{2}} \\ 4cy & = x^{2} \end{align}$$

$ $

===== Ellipse =====

  * 2 focal points

An ellipse is a locus of points such that the sum of $d_1$, the distance from one focus, and $d_2$, the distance from the other focus, is a constant.

$$d_1 + d_2 = 2a$$

===== Hyperbola =====

  * 2 focal points

A hyperbola is a locus of points such that the difference of $d_1$, the distance from one focus, and $d_2$, the distance from the other focus, is a constant.

$$|d_{2}-d_{1}|=2a$$