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. ($B=0$)

• 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.

$$d_1 = d_2$$

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

• 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.

$$d_1 + d_2 = 2a$$

• 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.

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