





Circle

Ellipse (h)

Parabola (h)

Hyperbola (h)

Definition:
A conic section is the intersection of a plane and a cone. 
Ellipse (v)

Parabola (v)

Hyperbola (v)

By changing the angle and location of intersection, we can produce a circle,
ellipse, parabola or hyperbola; or in the special case when the plane touches
the vertex: a point, line or 2 intersecting lines.
The General Equation for a Conic Section:
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 
The type of section can be found from the sign
of: B^{2}  4AC
If B^{2}  4AC is... 
then the curve is a...

< 0 
ellipse, circle, point or no curve.

= 0 
parabola, 2 parallel lines, 1 line or no curve.

> 0 
hyperbola or 2 intersecting lines.

The Conic Sections. For any of the below with a center (j, k)
instead of (0, 0), replace each x term with (xj) and each y
term with (yk).

Circle 
Ellipse 
Parabola 
Hyperbola 
Equation (horiz. vertex): 
x^{2} + y^{2} = r^{2} 
x^{2} / a^{2} + y^{2} / b^{2} =
1 
4px = y^{2} 
x^{2} / a^{2}  y^{2} / b^{2} =
1 
Equations of Asymptotes: 



y = ± (b/a)x 
Equation (vert. vertex): 
x^{2} + y^{2} = r^{2} 
y^{2} / a^{2} + x^{2} / b^{2} =
1 
4py = x^{2} 
y^{2} / a^{2}  x^{2} / b^{2} =
1 
Equations of Asymptotes: 



x = ± (b/a)y 
Variables: 
r = circle radius 
a = major radius (= 1/2 length major axis)
b = minor radius (= 1/2 length minor axis)
c = distance center to focus 
p = distance from vertex to focus (or directrix) 
a = 1/2 length major axis
b = 1/2 length minor axis
c = distance center to focus 
Eccentricity: 
0 
c/a 
1 
c/a 
Relation to Focus: 
p = 0 
a^{2}  b^{2} = c^{2} 
p = p 
a^{2} + b^{2} = c^{2} 
Definition: is the locus of all points which meet the condition... 
distance to the origin is constant 
sum of distances to each focus is constant 
distance to focus = distance to directrix 
difference between distances to each foci is constant 
Related Topics: 
Geometry section on Circles 





