MTH 108
Precalculus Mathematics II
9.1 Conics Notes
L. Sterling
Abstract
Provide a generalization to each of the key terms listed in this section.
Right Circular Cone
The collection of points that were generated thanks to the line, which is normally labeled g, that
also has an altitude intersects that plane of the circle at the circle’s center with their segments
that helps connect the base to the vertex form with the cone’s lateral surface.
Axis
The given [ﬁxed] line, which is normally labeled a, of a cone.
Vertex
The vertex, which is normally labeled V , is the given point, which is normally P, of a cone.
Generators
The given lines of a cone/conic section that are passing through the vertex, which is labeled V ,
and also making the same angle with the axis, which is still labeled a, and the line, which is still
labeled g.
Nappes
A cone’s two parts that are interesting the vertex, which is normally labeled V .
Conic Sections
The various curves that would be resulting from both the intersection of both a plane and the right
circular cone.
Circles
A conic that both does not contain a vertex on the place and also occurs when the plane would be
perpendicular to a cone’s axis while intersecting each generator.
Ellipses
A conic that both does not contain a vertex on the place and also occurs when the plane is being
tilted slightly so that it does intersect each of the generators, but only intersecting at only one of
the cone’s nappes.
1 Parabolas
A conic that both does not contain a vertex on the place and also occurs when the plane is being
tilted a little farther to make it parallel to only one generator while intersecting at only one of the
cone’s nappes.
Properties of Parabolas
Origin with Focus (a;0)
There is a parabolic equation with a vertex at the origin, which is normally (0;0), and with a focus,
which is normally labeled with F, at the point of (a;0). First, notice the vertex is at the origin
while the focus is at the given point of (a;0), which is then that the directix has the equation,
which is x = ▯a. If you let P(x;y) be a point on the parabola, them the following will occur [based
on using the distance formula]:
dist(P;F) = dist(P;Q)
q q
(x ▯ a) + (y ▯ 0) = (x + a) + (y ▯ y)
▯q ▯ 2 ▯ q ▯2
2 2 2 2
(x ▯ a) + (y ▯ 0) = (x + a) + (y ▯ y)
(x ▯ a) + (y ▯ 0) = (x + a) + (y ▯ y)
2 2 2 2
(x ▯ a) + (y) = (x + a) + (0)

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