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Conic Sections Cheat Sheet

A Education › Mathematics cheat sheet in English by CROSSANT

Graphics sourced from these websites: Conic Cross-Sections https://www.ck12.org/book/ck-12-algebra-ii-with-trigonometry-concepts/section/10.0/ Labelled Parabola, Ellipse, and Hyperbola https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_%28Apex%29/09%3A_Curves_in_the_Plane/9.01%3A_Conic_Sections

Calculus K Focus H Conics Parabolas Hyperbola
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Parabolas with vertex (h,k)

Opening up/down
(x-h)2 = ±4p(y-k)
Vertical Focus
(h, k+p)
Directrix
y=k-p
Opening right/left
(y-k)2 = ±4p(x-h)
Horizontal Focus
(h+p, k)
Directrix
x = h-p
Any point on a parabola is equidi­stant from the parabo­la's focus and directrix

Conic Cross-­Sec­tions Diagram

Conic Cross-Sections Diagram

Parabola opening upwards

Parabola opening upwards

Circle­s/E­llipses with center (h,k)

Circle
(x-h)2 + (y-k)2 = r2
Circle Focus
(h,k)
Circle Vertices
None
Wide Ellipse
(x-h)2/a2 + (y-k)2/b2 = 1
Wide Foci
(h±c, k)
Wide Vertices
(h±a, k±b)
Tall Ellipse
(x-h)2/b2 + (y-k)2/a2 = 1
Tall Foci
(h, k±c)
Tall Vertices
(h±b, k±a)
c² = a² - b² and a > b > 0
Formulas for foci generate two different points (+c and -c), and formulas for vertices generate four different vertices: (h+a, k) (h-a, k) (h, k+b) and (h, k-b)
Distances between a focal point to any point on the ellipse, plus the distance of the other focal point to that same point on the ellipse, gives a sum of distances that is constant for any point on the ellipse

Wide Ellipse

Wide Ellipse

Hyperbolas with center (h,k)

Branches opening horizo­ntally
(x-h)2/a2 - (y-k)2/b2 = 1
Horizontal Foci
(h±c, k)
Horizontal Vertices
(h±a, k)
Asymptotes
y-k = ±(b/a)­(x-h)
Branches opening vertically
(y-k)2/a2 - (x-h)2/b2 = 1
Vertical Foci
(h, k±c)
Vertical Vertices
(h, k±a)
Asymptotes
y-k=±(­a/b­)(x-h)
c² = a² + b² and a, b ≠ 0
Formulas for foci generate two different points (+c and -c), formulas for vertices generate two different points (+a and -a), and formulas for asymptotes generate two different asymptotes (+(a/b) and -(a/b) or +(b/a) and -(b/a))
Distance of a focal point to a point on either hyperbola branch, minus distance of the other focal point to that same point on that same hyperbola branch, gives a value whose magnitude is constant for any point on either hyperbola branch

Horizontal Hyperbola

Horizontal Hyperbola

Horizontal Hyperbola Asymptotes

Horizontal Hyperbola Asymptotes

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