Conics Formula Sheet

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Conics Summary
Conic Section
Standard Form
Other Info.
Derived from the distance formula.
Circle
2
2
2
(
x
h
)
(
y
k
)
r
( k
h
,
)
Centre
Radius r
2
2
(
x
x
)
(
y
y
)
d
2
1
2
1
( k
h
,
)
p
0
opens
up,
Parabola - Vertex
2
(
x
h
)
4
p
(
y
k
)
(
h
,
k
p
)
Focus
p
0
opens
down
y 
k
-
p
Directrix at
p
0
opens
right,
h 
(
p
,
k
)
Foci
2
(
y
k
)
4
p
(
x
h
)
p
0
opens
left
x 
h
p -
Directrix at
The longer axis is called the major
( k
h
,
)
Ellipse - Centre
axis, the shorter axis is called the
2
2
(
x
h
)
(
y
k
)
minor axis.
1
Horizontal major axis: a > b
2
2
a
b
h 
(
a
,
k
)
Vertices:
‘a’ is the distance from the centre
h 
to each vertex (the end of the
(
c
,
k
)
Foci:
major axis).
‘b’ is the distance from the centre
to the end of the minor axis.
2
2
(
y
k
)
(
x
h
)
1
Vertical major axis: a > b
‘c’ is the distance from the centre
2
2
a
b
to each focus.
2
2
2
Vertices:
(
h
,
k
a
)
c
a
b
Foci:
(
h
,
k
c
)
Length of major axis = 2a
Length of minor axis = 2b
‘a’ is the distance from the centre
( k
h
,
)
Hyperbola - Centre
to each vertex.
2
2
(
x
h
)
(
y
k
)
Horizontal transverse axis
‘b’ is a point on the conjugate axis
1
(x coefficient is positive)
2
2
but is not a point on the hyperbola
a
b
h 
Vertices:
(it helps determine asymptotes)
(
a
,
k
)
h 
Foci:
(
c
,
k
)
‘c’ is the distance from the centre
to each focus.
2
2
2
c
a
b
Vertical transverse axis
(y coefficient is positive)
2
2
Note: The transverse axis is not
(
y
k
)
(
x
h
)
Vertices:
(
h
,
k
a
)
1
necessarily the longer axis but is
2
2
Foci:
(
h
,
k
c
)
associated with whichever variable
a
b
is positive
If A C 
If A C 
If A C 
2
2
*
0
*
0
*
0
If A C
Ax
Cy
Dx Ey F
0
ELLIPSE
HYPERBOLA
PARABOLA
CIRCLE

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