Conic Sections
Parabola
Ellipse
Parabola
A set of points in
from fixed point F
called the Directrix
a
Hyperbola
plane that
called a Focus
are
equidistant
and a fixed line
Turning point called vertex
Deriving equation
For
y
open
vertex
p
n
p
y
p
up y k
a
h
aply k
y k
Focus
directrix
h
shyfts
h
4p a h
k
Directrix
h
p
K P
y 14 up a
Focus h p k
Directrix
x
h p
Notes
mm
V
pro
n
pro
Ak
y upx
Noteim
If only
Similarly proven
a
vy is squared then parabola
LI
n.T
all
mo
of symmetry
yess
Meggie
a
of symmetry
Etyn
anonymity
if
t.is
tIii1
if
Gas
Ellipses
An ellipse is a set of points in the plane
2 fixed points Fi n Fe is
distances from
2 Fixed points called Foci
Play
b
get
co
vertical
1
PF.lt PF2
3
From
with origin
29
Δ F EP
229
2
C Ca
4 let
semi major axis major radius
b
semi minor axis minor radius
midpt 0,0
c o
y axis
c
ca
a
c
2
azb
a2 b
c
c
a
n
b
an
o
1a o
vertices
vertices
a'b
1
azbso
1
where
ay
x
22
1
where
so
at c
b
then b
a
Foci on x axis
C
3
minor axis skinny
foci o
x axis
in middle
major axis bree
Foci on
of whose
constant
a
foci on
1 Place
b
foci
sum
Steps
a
a
the
o
Ial
Hyperbola
The set of all points
distances
from
in a
the
plane
fixed points Fi n Fz
2
difference of whose
the foci is a constant
PF PFal 129
P
ny.fr
s
Eht
a
play
Tt
radio
where
ca4b
vertices
19,0
Foci
C 0
Note when
drawing
22
Wen acc
but always under
0
doe one with a
is
term
TE
Y
4 1
major access
gpoint
23
Foci
2
start with asymtotes
faint
c c
vertices
Asymptotes
Nue if
where
a
lab
z
b
o
az b
ME
a
y
a
9
1
h
y k
assymtotes will
a
Foci
o
c
where
vertices
o
a
asymptotes
y
intersect
h k
nv
c
b
In
be
are