Cheatography

# Conic Sections Cheat Sheet 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

### 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

### 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

### 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