Math 30P     Conic(Sections)     Conic(General)     Conic(Standard)      Prerequisite Skills

Conic Sections

conic section: a curve produced when a plane intersects a right-circular cone

degenerate conic section: a point, line, or pair of lines that arise as limiting forms of a conic

Double-napped Cone

double-napped cone: two identical but opposite cones that share a common vertex

central axis: the axis of symmetry of a right-circular cone

vertex: the point at which the generator is rotated to create the cones

generator: a triangle/line that sweeps out a surface, such as a cylinder or a right-circular cone, when rotated about a central axis

Cylinder

A cylinder can be thought of as a special case of the cone. Remember, the cone is formed by rotating a generator about the central axis. If you change the orientation of the generator so the angle between the generator and the central axis decreases until it is parallel to the central axis, the cone becomes a cylinder.

 

Conic Section and the Generator Angle

Animation: http://www.math.odu.edu/cbii/calcanim/consec.avi

generator angle = 20o

generator angle = 50o

 

(x-h)2+(y-k)2=r2

 

 

Intersecting (cutting) plane angle = 90o

When a double-napped cone is intersected by a plane at a right angle to its axis, the cross section is a circle.

Intersecting (cutting) plane angle = 90o
Intersecting (cutting) plane angle = 90o

 

x = a(x - k)2 + h
y = a(x - h)2 + k
Ax2 + Cy2 + Dx + Ey + F = 0

 

 

Intersecting (cutting) plane angle (θ) = generator angle

Intersecting(cutting) plane is parallel (equal) to the generator

If the plane intersects the double-napped cone parallel to a generator, the cross section is a parabola.

If generator angle = 20o

θ = Intersecting (cutting) plane angle = 20o

If generator angle = 50o

θ = Intersecting (cutting) plane angle = 50o

 

(x-h)2+(y-k)2=1
  a2        b2

 

 

Intersecting (cutting) plane angle (θ):

generator angle < θ < 90o

If the plane intersects one nappe of a double-napped cone at neither a right angle to the axis nor parallel to a generator, then the cross section is an ellipse.

If generator angle = 20o

Intersecting (cutting) plane angle (θ):

20o < θ < 90o

If generator angle = 50o

Intersecting (cutting) plane angle (θ):

50o < θ < 90o

 

(x - h)2 - (y - k)2 = +1
   a2           b2

 

 

Intersecting (cutting) plane angle (θ):

0 < θ < generator angle

When both nappes of a double-napped cone are intersected by a plane (not passing through the vertex), the cross section produces a hyperbola.

If generator angle = 20o

Intersecting (cutting) plane angle (θ):

0 < θ < 20o

If generator angle = 50o

Intersecting (cutting) plane angle (θ):

0 < θ < 50o

Note: Degenerate cases can be created using a plane and double-napped cones and/or cylinder.

Conic (Degenerate Conic) Description

circle (point)

Degenerate Cases:
if r = 0, the graph is one point

if r < 0, there is no graph

ellipse (point)

Degenerate Cases:

(x-h)2+(y-k)2=0, the graph is one point
  a2        b2

(x-h)2+(y-k)2=-1, there is no graph
  a2        b2

parabola (line)

line

Degenerate Cases:

A = C = 0, line (above)

Ax2 + Dx + F = 0, line (above), 2 parallel lines, no graph

Cy2 + Ey + F = 0, line (above), 2 parallel lines, no graph

hyperbola (two intersecting lines)

Degenerate Cases:

(x-h)2-(y-k)2 = 0, two intersecting lines
   a2       b2

 

Note: Degenerate cases can be created using a plane anddouble-napped cones and/or cylinder.
Conic (Degenerate Conic) Description

circle (point)

If a plane slices the cylinder at a right angle to the central axis (as for the cone), a circle is produced. If the cylinder radius = 0, the circle will become a point.

ellipse (point)

If the plane cuts the cylinder at any angle less than 90°, an ellipse is produced. If the cylinder radius = 0, the ellipse will become a point.

(2 parallel lines)

If the plane cuts into the cylinder, it will produce two parallel lines. It can't create with a double-napped cone and a plane.

(no graph)

If the plane doesn’t touch the cylinder at all, it will produce no graph. It can't create with a double-napped cone and a plane..

Note: Degenerate cases can be created using a plane and double-napped cones and/or cylinder.

Conic (Degenerate Case) Double-napped Cone Cylinder

(point)

diameter of cylinder = 0

(line)

 

If the plane touches at the edge pf the cylinder, it will produce one line.

(2 intersecting lines)

It can't create with a cylinder and a plane.

Note: the cutting plane must pass through the vertex of the double-napped cone at an angle, θ, with the central axis within 0 < θ < 52.5o.

(2 parallel lines)

2 parallel lines can't be created with a double-napped cone and a plane.

(no graph)

No graph can't be created with a double-napped cone and a plane.

 

This work has been adapted from a Math 30 Pure learning resource originally produced and owned by Alberta Education

 

Math 30P     Conic(Sections)     Conic(General)     Conic(Standard)      Prerequisite Skills

 

Comments to:  Jim Reed - Homepage
Started September, 1998. Copyright © 2006, 2007, 2008