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Lecture 24

# MTH 108 Lecture 24: 9.1 Conics Notes Premium

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School
Department
Mathematics
Course
MTH 108
Professor
Lun- Yi Tsai
Semester
Fall

Description
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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