1 conics

1.1 ellipse

Described by a set of points in the plane for which the sum of distances from two fixed points (called foci F, F') is constant.

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Foci are located at:

\begin{cases} (\pm c, 0) & \text{if} & a > b \\ (0, \pm c) & \text{if} & a < b \\ \end{cases}

Where c = \sqrt{|a^2 - b^2|}

1.2 hyperbola

Described by a set of points in the plane for which the difference of distances from two fixed points (called foci F, F') is constant.

Let c = \sqrt{a^2 + b^2}

1.2.1 Foci on x axis

1.2.1.1 equation

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

1.2.1.2 foci

(\pm c, 0)

1.2.2 Foci on y axis

1.2.2.1 equation

\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1

1.2.2.2 foci

(0, \pm c)

1.3 parabola

Described by a set of points in the plane for which the distances from a fixed point (called focus) and a fixed line (called the directrix) are equal.

1.3.1 Focus on x axis

x = \frac{1}{4c}y^2

F = (c, 0)

Directrix: x = -c

1.3.2 Focus on y axis

y = \frac{1}{4c}x^2

F = (0, c)

Directrix: y = -c