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AP Calculus BC 9.1 Conics
By changing the angle and location of
intersection, we can produce a circle,
ellipse, parabola, and hyperbola.
In the special case when the plane
touches the vertex, a point, line, or 2
intersecting lines result.
The General Equation for a Conic Section:
��� � ��� � ��� � �� � � � � 0
AP Calculus BC 9.1 Conics
Let c represent the center-to-focus distance.
The Conic Cente
r Equation Vertex Focus
(Foci)
Other Locus is set of
all points
such that:
Parabola
(vertical axis
of symmetry)
�� � 4��
(0,0)
(0, �)
Directrix at
� � −�
Distance to
focus =
Distance to
Directrix
Parabola
(horizontal
axis of
symmetry)
�� � 4��
(0,0)
(�, 0)
Directrix at
� � −�
Parabola
(vertical axis
of symmetry)
(� − ℎ)� � 4�(� − �)
(ℎ, �)
(ℎ, � � �)
Directrix at
� � � − �
Parabola
(horizontal
axis of
symmetry)
(� − �)� � 4�(� − ℎ)
(ℎ, �)
(ℎ � �, �)
Directrix at
� � ℎ − �
Ellipse
(horizontal
major axis)
(0,0) ��
�����
��� 1
(±�, 0) (±�, 0) �� � �� − �� Sum of
Distance from
the point to
each foci is
constant
Ellipse
(vertical
major axis)
(0,0) ��
�����
��� 1
(0, ±�) (0,±�) �� � �� − ��
Ellipse
(horizontal)
(ℎ, �) (� − ℎ)�
���(� − �)�
��� 1
(ℎ ± �, �) (ℎ ± �, �) �� � �� − ��
Ellipse
(vertical)
(ℎ, �) (� − ℎ)�
���(� − �)�
��� 1
(ℎ, � ± �) (ℎ, � ± �) �� � �� − ��
Hyperbola
(horizontal
major axis)
(0,0) ��
��−��
��� 1
(±�, 0) (±�, 0) �� � �� � ��
Asymptotes:
� � ±�
��
Difference of
Distance from
a point to
each foci is
constant
Hyperbola
(vertical major
axis)
(0,0) ��
��−��
��� 1
(0, ±�) (0,±�) �� � �� � ��
Asymptotes:
� � ±�
��
Hyperbola
(horizontal
major axis)
(ℎ, �) (� − ℎ)�
��−(� − �)�
��� 1
(ℎ ± �, �) (ℎ ± �, �) �� � �� � ��
Asymptotes:
� − �
� ±�
�(� − ℎ)
Hyperbola
(vertical major
axis)
(ℎ, �) (� − �)�
��−(� − ℎ)�
��� 1
(ℎ, � ± �) (ℎ, � ± �) �� � �� � ��
Asymptotes:
� − �
� ±�
�(� − ℎ)
Eccentricity of an ellipse or hyperbola: � ��
�
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