Unit 2
Conics
The Cone and Conic Sections
In this unit we will be dealing with sections of
a cone.
Shapes such as circles, ellipses, hyperbolas,
and parabolas are called conic sections
because they can all be found using a cone.
If we take two cones and place their two vertices
together, we get a double cone or double-napped
cone. The sides of a cone are now called the
generator. The middle point is the vertex and the
open circular ends are the bases. Down the
middle of the double-napped cone is the central
axis, it is perpendicular to the base.
• The most common type of cone is the right
circular cone. Its base is a circle. If we
slice a right circular cone at different
angles, the circle, ellipse, parabola, and the
hyperbola are generated.
If a single cone is cut parallel to the base,
what shape do we get?
A circle is formed when a plane intersects
either nappe, parallel to the base.
If a single cone is cut inclined to the base,
what shape do we get?
An ellipse is formed when a plane intersects
either nappe, not perpendicular to the
central axis and less than parallel to the
generator
If a double-napped cone is cut parallel to the
central axis, what shape do we get?
A hyperbola is formed when a plane
intersects both nappes of a cone.
If a single cone is cut parallel to the cone’s
generator, what shape do we get?
A parabola is formed when a plane intersects
one nappe, parallel to the generator.
Identify two characteristics of each conic
section.
Which conic sections are symmetrical? How
are they symmetrical?
Which of the conic sections are closed
figures? Open figures?
Compare and contrast the ellipse and the
circle.
Compare and contrast the parabola and the
hyperbola.
Some slices of a double-napped cone cut by a
plane do not yield a circle, an ellipse, a
parabola, or a hyperbola. These are called
degenerate conic sections.
Circle/Ellipse
Each point gets smaller as the plane slides
through the core to the vertex. At the
vertex, the circle or ellipse becomes a point.
Parabola
A parabola is a curved line. It gets narrower
as it approaches the generator. At the
generator, it becomes a straight line.
Hyperbola
A hyperbola is two curves. The curves approach one
another as the plane nears the vertex. At the
vertex, they cross and become an X.
General Form of a Conic
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 where A,B, and
C are not all 0
For our purposes B always equals 0 so that
the axes of symmetry will always be the
horizontal x-axis and the vertical y-axis. In
other words, the axes of the conic section
are parallel to the x- and/or y-axes.
Therefore, the new General Form is:
2
2
Ax + Cy + Dx + Ey + F = 0
PROPERTIES
A and C are never both 0 as there has to be at
least one squared term for the general form
to hold true.
If A= C the graph may be a circle
If A does not equal C the graph may be an
ellipse
If AC = 0 the graph may be a parabola
IF AC < 0 the graph may be a hyperbola
Why can’t A, B, and C not all equal 0?