Cheat Sheet: Conic sections
Parabolas in standard position: vertex = (0,0)
y 2 = 4 px
focus: (p,0)
directrix: x=-p
y 2 = −4 px
focus: (-p,0)
directrix: x=p
x 2 = 4 py
focus: (0,p)
directrix: y=-p
x 2 = −4 py
focus: (0,-p)
directrix: y=p
Ellipses in standard position: center = (0,0)
semimajor axis=a 
 if a=b we have a "degenerate" ellipse, better known as a circle
semiminor axis=b 
distance from center to focus = c =
x2 y 2
+
=1
a 2 b2
vertices: (a,0) and (-a,0)
foci: (c,0) and (-c,0)
a 2 − b2
x2 y2
+
=1
b2 a 2
vertices: (0,a) and (0, -a)
foci: (0,c) and (0,-c)
Hyperbolas in standard position: center is (0,0)
semifocal axis=a
semiconjugate axis = b
distance from center to focus = c =
x2 y 2
−
=1
a 2 b2
vertices: (a,0) and (-a,0)
foci: (c,0) and (-c,0)
b
asymptotes: y = ± x
a
a 2 + b2
y 2 x2
− =1
a2 b2
vertices: (0,a) and (0,-a)
foci: (0,c) and (0,-c)
a
asymptotes: y = ± x
b
Parabolas in translated position: vertex = (h,k) (pictures drawn with h and k both positive)
( y − k ) 2 = 4 p ( x − h)
focus: (h+p,k)
directrix: x=h-p
(y-k)2 = −4 p ( x − h)
focus: (h-p,k)
directrix: x=h+p
(x-h)2 = 4 p ( y − k )
focus: (h,k+p)
directrix: y=k-p
(x-h)2 = −4 p ( y − k )
focus: (h,k-p)
directrix: y=k+p
Ellipses in translated position: center = (h,k) (again, pictures drawn for h and k positive)
semimajor axis=a 
 if a=b we have a "degenerate" ellipse, better known as a circle
semiminor axis=b 
distance from center to focus = c =
a 2 − b2
( x − h) 2 ( y − k ) 2
+
=1
a2
b2
vertices: (h+a,k) and (h-a,k)
foci: (h+c,k) and (h-c,k)
(x-h) 2 ( y − k ) 2
+
=1
b2
a2
vertices: (h,k+a) and (h, k-a)
foci: (h,k+c) and (h,k-c)
Hyperbolas in translated position: center is (h,k) (again, pictures drawn for h, k positive)
semifocal axis=a
semiconjugate axis = b
distance from center to focus = c =
( x − h) 2 ( y − k ) 2
−
=1
a2
b2
vertices: (h+a,k) and (h-a,k)
foci: (h+c,k) and (h-c,k)
b
asymptotes: y − k = ± ( x − h)
a
a 2 + b2
( y − k ) 2 ( x − h) 2
−
=1
a2
b2
vertices: (h,k+a) and (h,k-a)
foci: (h,k+c) and (h,k-c)
a
asymptotes: y − k = ± ( x − h)
b
Rotations of conic sections:
An equation of the form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 is the equation representing a conic
section. If B is not 0, view this as a rotated conic section. To graph:
A−C
 π
1. Find the angle θ in  0,  such that cot(2θ )=
. Rotate the x and y axes θ to
B
 2
create a new pair of axes we will call the x’ and y’ axes.
2. Substitute:
x=x’cos θ -y’sin θ
y=x’sin θ +y’cos θ
to get a new equation describing the conic section in terms of x’ and y’.
3. Graph the equation from step 2 on the x’ and y’ axes.
Conic sections in polar coordinates:
In polar coordinates, we describe all non-degenerate conic sections in relation to a fixed point
(called the focus ) and a fixed line (called the directrix)
If we let the focus be the pole, and the directrix be a line that is either parallel or perpendicular to
the polar axis, ALL conic sections can be described using one of the following 4 types of
equations, with the focus and directrix as shown:
ed
1 + e cos θ
directrix is d units
ed
1- e cos θ
directrix is d units
ed
1 + e sin θ
directrix is d units
ed
1 − e sin θ
directrix is d units
right of the pole
left of the pole
above the pole
below the pole
r=
r=
r=
r=
e is called the eccentricity and tells us what type of conic section we have:
e=1
0<e<1
e>1
parabola
ellipse (this does not include circles—circles count as “degenerate” ellipses)
hyperbola
To graph, we use the equation in polar coordinates to determine the important values for the
equation in rectangular coordinates. We’ll go back and forth between the two types of
coordinates by working on a graph in which the pole (for polar coordinates) coincides with the
origin (for rectangular coordinates) and the polar axis (for polar coordinates) coincides with the
positive x axis (for rectangular coordinates):
Parabolas: need to know vertex and “p”
vertex must be the point half-way between the focus and directrix
p=distance from focus to vertex=distance from directrix to vertex= ½ d
Ellipses: need to know center, “a,” “b,” and “c” (c is needed in order to find the other values)
1. The vertices will lie on either the x or y axis—whichever is perpendicular to the
directrix. Since one vertex will lie on either side of the focus, find them by plugging in
0 and π if the vertices lie on the x-axis

appropriate values for θ : π
π
 2 and - 2 if the vertices lie on the y-axis
2. Plot the vertices. The center must be ½ -way in between them. Find a, b, and c
a = distance from center to vertex
c = distance from center to focus
b= a 2 − c 2
3. Sketch.
Hyperbolas: need to know center, “a,” “b,” and “c” (c is needed in order to find the other values)
1. The vertices will lie on either the x or y axis—whichever is perpendicular to the
directrix. Since one vertex will lie on either side of the focus, find them by plugging in
0 and π if the vertices lie on the x-axis

appropriate values for θ : π
π
 2 and - 2 if the vertices lie on the y-axis
2. Plot the vertices. The center must be ½ -way in between them. Find a, b, and c
a = distance from center to vertex
c = distance from center to focus
b= c 2 − a 2
3. Sketch.