Conic Sections

General Equation: Circle

(x-h)²+(y-k)²=r²

General Equation: Ellipse, Horizontal

(x-h)²/a²+(y-k)²/b²=1

General Equation: Ellipse, Vertical

(x-h)²/b²+(y-k)²/a²=1

Center: All Conics

(h,k)

Vertices: Ellipse, Horizontal

(h±a, k)

Vertices: Ellipse, Vertical

(h,k±b)

Covertices: Ellipse, Horizontal

(h,k±b)

Covertices: Ellipse, Vertical

(h±b, k)

Foci: Ellipse, Horizontal

(h±c, k) where c² = a² - b², a>b

Foci: Ellipse, Vertical

(h, k±c) where c² = a² - b², a>b

Length of Major Axis: Ellipse

2a

Length of Minor Axis: Ellipse

2b

General Equation: Hyperbola, Left/right

(x-h)²/a² - (y-k)²/b² = 1

General Equation: Hyperbola, Up/down

(y-k)²/a² - (x-h)²/b² = 1

Vertices: Hyperbola, Left/right

(h±a, k).

Vertices: Hyperbola, Up/down

(h, k±a).

Foci: Hyperbola, Left/right

(h±c, k), where c = √(a² + b²).

Foci: Hyperbola, Up/down

(h, k±c), where c = √(a² + b²).

Slant Asymptotes: Hyperbola, Left/right

y = k ± (b/a)(x - h).

Slant Asymptotes: Hyperbola, Up/down

y = k ± (a/b)(x - h).

Standard Form: Parabola, Up

(x-h)²=4p(y-k)

Standard Form: Parabola, Down

(x-h)² = -4p(y-k)

Standard Form: Parabola, Left

(y-k)² = -4p(x-h)

Standard Form: Parabola, Right

(y-k)² = 4p(x-h)

Focus: Parabola, Up

(h, k+p)

Focus: Parabola, Down

(h, k-p)

Focus: Parabola, Right

(h+p, k)

Focus: Parabola, Left

(h-p, k)

Directrix: Parabola, Up

y = k-p

Directrix: Parabola, Down

y = k+p

Directrix: Parabola, Left

x = h+p

Directrix: Parabola, Right

x = h-p