Show Menu
Cheatography

Conic Sections Cheat Sheet by

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

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

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

 

Hyperbolas with center (h,k)

Pair opening left and right
(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)
Pair opening up and down
(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, |a|≠0, |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 pair of Hyperbolas

Horizontal Hyperbola Asymptotes

                       
 

Comments

No comments yet. Add yours below!

Add a Comment

Your Comment

Please enter your name.

    Please enter your email address

      Please enter your Comment.

          Related Cheat Sheets

          Sequences and Series Cheat Sheet
          Calculus II Cheat Sheet

          More Cheat Sheets by CROSSANT

          Calculus II Cheat Sheet
          Integral Trigonometry Cheat Sheet