Math 30P Conic(Sections) Conic(General) Conic(Standard) Prerequisite Skills
conic section: a curve produced when a plane intersects a rightcircular cone
degenerate conic section: a point, line, or pair of lines that arise as limiting forms of a conic
doublenapped cone: two identical but opposite cones that share a common vertex
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 rightcircular 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. 
generator angle = 20^{o } 
generator angle = 50^{o} 

(xh)^{2}+(yk)^{2}=r^{2}
Intersecting (cutting) plane angle = 90^{o} When a doublenapped cone is intersected by a plane at a right angle to its axis, the cross section is a circle. 
Intersecting (cutting) plane
angle = 90^{o} 
Intersecting (cutting) plane
angle = 90^{o} 
x = a(x  k)^{2} +
h
Intersecting (cutting) plane angle (θ) = generator angle Intersecting(cutting) plane is parallel (equal) to the generator If the plane intersects the doublenapped cone parallel to a generator, the cross section is a parabola. 
If generator angle = 20^{o} θ = Intersecting (cutting) plane angle = 20^{o } 
If generator angle = 50^{o} θ = Intersecting (cutting) plane angle = 50^{o } 
(xh)^{2}+(yk)^{2}=1^{}
Intersecting (cutting) plane angle (θ): generator angle < θ < 90^{o} If the plane intersects one nappe of a doublenapped cone at neither a right angle to the axis nor parallel to a generator, then the cross section is an ellipse. 
If generator angle = 20^{o} Intersecting (cutting) plane angle (θ): 20^{o} < θ < 90^{o} 
If generator angle = 50^{o} Intersecting (cutting) plane angle (θ): 50^{o} < θ < 90^{o} 
(x  h)^{2}  (y
 k)^{2} = +1
Intersecting (cutting) plane angle (θ): 0 < θ < generator angle When both nappes of a doublenapped cone are intersected by a plane (not passing through the vertex), the cross section produces a hyperbola. 
If generator angle = 20^{o} Intersecting (cutting) plane angle (θ): 0 < θ < 20^{o} 
If generator angle = 50^{o} Intersecting (cutting) plane angle (θ): 0 < θ < 50^{o} 
Note: Degenerate cases can be created using a plane and doublenapped 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: (xh)^{2}+(yk)^{2}=0, the graph is one
point (xh)^{2}+(yk)^{2}=1, there is no
graph 
parabola (line) line 
Degenerate Cases: A = C = 0, line (above) Ax^{2 }+ Dx + F = 0, line (above), 2 parallel lines, no graph Cy^{2 }+ Ey + F = 0, line (above), 2 parallel lines, no graph 
hyperbola (two intersecting lines) 
Degenerate Cases: (xh)^{2}(yk)^{2} = 0, two
intersecting lines 
Note: Degenerate cases can be created using a plane anddoublenapped cones and/or cylinder.
Note: Degenerate cases can be created using a plane and doublenapped cones and/or cylinder.
This work has been adapted from a Math 30 Pure learning resource originally produced and owned by Alberta Education