The Conics

By: Diana Brown


Day Ten:

Summary


Conic Sections

circle conic

ellipse conic

parabola conic

hyperbola conic

Circle
graph circle (horiz.)

Ellipse (h)
graph ellipse (horiz.)

Parabola (h)
graph parabola (horiz.)

Hyperbola (h)
graph hyperbola (horiz.)

Definition:
A conic section is the intersection of a plane and a cone.

Ellipse (v)
graph ellipse (vert.)

Parabola (v)
graph parabola (vert.)

Hyperbola (v)
graph hyperbola (vert.)


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.

 

point conic

line conic

double line conic

 

Point
graph point conic

Line
graph line conic

Double Line

 

The General Equation for a Conic Section:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

The type of section can be found from the sign of: B2 - 4AC

If B2 - 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 (x-j) and each y term with (y-k).

 

Circle

Ellipse

Parabola

Hyperbola

Equation (horiz. vertex):

x2 + y2 = r2

x2 / a2 + y2 / b2 = 1

4px = y2

x2 / a2 - y2 / b2 = 1

Equations of Asymptotes:

 

 

 

y = ± (b/a)x

Equation (vert. vertex):

x2 + y2 = r2

y2 / a2 + x2 / b2 = 1

4py = x2

y2 / a2 - x2 / b2 = 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

c/a

Relation to Focus:

p = 0

a2 - b2 = c2

p = p

a2 + b2 = c2

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

 


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