Horizontal ellipse, center at origin

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

Vertical ellipse, center at origin

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

Horizontal ellipse, center

(h, k)

\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1, a > b

Vertical ellipse, center

(h, k)

\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1, a > b

Hyperbola, center at origin, transverse axis on x-axis

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

Hyperbola, center at origin, transverse axis on y-axis

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

Hyperbola, center at

(h, k),

transverse axis parallel to x-axis

\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1

Hyperbola, center at

(h, k),

transverse axis parallel to y-axis

\frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1

Parabola, vertex at origin, axis of symmetry on x-axis

y^{2} = 4 p x

Parabola, vertex at origin, axis of symmetry on y-axis

x^{2} = 4 p y

Parabola, vertex at

(h, k),

axis of symmetry on x-axis

{(y - k)}^{2} = 4 p (x - h)

Parabola, vertex at

(h, k),

axis of symmetry on y-axis

{(x - h)}^{2} = 4 p (y - k)

General Form equation of a conic section

A x^{2} + B x y + C y^{2} + D x + E y + F = 0

Rotation of a conic section

\begin{array}{l} x = x^{'} \cos θ - y^{'} \sin θ \\ y = x^{'} \sin θ + y^{'} \cos θ \end{array}

Angle of rotation

θ, where \cot (2 θ) = \frac{A - C}{B}