Conic Sections
Circle
Standard Form
x  h2   y  k 2  r 2
Center
Radius
(h , k)
r
Parabola
Vertical Axis
Horizontal axis
Standard Form
x  h2  4 p y  k 
 y  k 2  4 px  h
Opens
Vertex
Focus
Directrix
Axis of symmetry
Up or down
(h , k)
(h , k + p)
y=k  p
x=h
Left or right
(h , k)
(h + p , k)
x=h  p
y=k
Focal length
Focal width (diameter)
P
| 4p |
P
| 4p |
Horizontal Major Axis
Vertical Major axis
Center
Focal axis
Foci
(h , k)
y=k
(h ± c, k)
(h , k)
x=h
(h, k ± c)
Vertices
Co-Vertices
Major axis
Minor axis
Semimajor axis
Semiminor axis
(h ± a, k)
(h, k ± b)
2a = length of major axis
2b = length of minor axis
a
b
(h, k ± a)
(h ± b, k)
2a = length of major axis
2b = length of minor axis
a
b
C (Pythagorean relation)
Eccentricity of an ellipse
a2 = b2 + c2 or c2 = a2
e = (c/a)
Ellipse
Standard Form
 b2
Hyperbola
Horizontal Transverse axis
Equation
x  h 
2
a2
y  k 
2

b2
1
Vertical Transverse Axis
 y  k 2  x  h2
a2
b2
Center
Focal axis
Foci
Vertices
Semitranverse axis
(h, k)
y=k
(h ± c, k)
(h ± a, k)
a
(h, k)
x=h
(h, k ± c)
(h, k ± a)
a
Semiconjugate axis
C (Pythagorean relation)
b
c2 = a2 + b2
b
c2 = a2 + b2
Asymptotes
yk
b
x  h 
a
yk
a
x  h 
b
1
Classifying conic
sections
General form:
Ax2+Cy2+Dx+Ey+F=0
Circles
Parabola
Ellipse
Hyperbola
A=C
AC=0, Both
are not 0
AC>0
AC<0
Discriminant test:
The second degree equation: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 graphs as
 A hyperbola if B2 – 4AC > 0
 A parabola if B2 – 4AC = 0
 An ellipse if B2 – 4AC < 0
Translation-of-Axes Formulas:
The coordinates (x, y) and (x’, y’) based on parallel sets of axes are related by either of the following translation formulas:
x = x’ + h and y = y’ + k
or
x’ = x – h and y’ = y – k
Rotation-of-Axes Formula:
The coordinates (x, y) and (x’, y’) based on rotated sets of axes are related by either of the following rotation formulas:
x’ = x cos + y sin and y’ = - x sin + y cos
or
x = x’ cos
where , 0 <
< , is the angle of rotation
Coefficients for a conic in a rotated system:
If we apply the rotation formulas to the general second-degree equation in x and y, we obtain a second-degree equation in x’
and y’ of the form2
A’x’2 + B’x’y’ + C’y’2 + D’x’ + E’y’ + F = 0
Where coefficients are
A’ = A cos2
B’ = B cos 2
C’ = C cos2
D’ = D cos + E sin
E’ = E cos
F’ = F
Angle of Rotation to Eliminate the Cross-Product Term
If B ≠ 0, an angle of rotation such that
cot 2
will eliminate the term B’x’y’ from the second-degree equation in the rotated x’y-coordinate system.