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: � ��

�