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# Chapter 11: Analytic Geometry MAT 109

Barry University

GPA 3.7

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This 16 page Bundle was uploaded by Sterling on Wednesday September 14, 2016. The Bundle belongs to MAT 109 at Barry University taught by Dr. Singh in Fall 2016. Since its upload, it has received 27 views. For similar materials see Precalculus Mathematics 1 in Mathmatics at Barry University.

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Date Created: 09/14/16

MAT 109 Precalculus Mathematics 1 11.1 Conics Notes L. Sterling September 19th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 (Right Circular) Cone The given collection of points that were generated thanks to the line, which is g. 2 Axis The given [▯xed] line, which is a, of a cone. 3 Vertex The given point, which is V , of a cone. 1 4 Generators The given lines of a cone/conic section that are passing through the vertex, which is V , and also making the same angle with the axis, which is a, and the line, which is g. 5 Nappes A cones two parts that are intersecting at V , which is the vertex. 6 Conics [Conic Sections] The curves that would be resulting from both the intersection of both a plane and the right circular cone. 7 Circles A conic that both does not contain a vertex on the place and also occurs when the plane would be perpendicular to a cones axis while intersecting each gener- ator. 8 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 cones nappes. 2 9 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 cones nappes. 10 Hyperbola A conic that both does not contain a vertex on the place and also occurs when the plane’s intersecting at both of the cones nappes. 11 Degenerate Conics A conic that occurs when a plane does not have vertex, which is the planes intersection and the cone is either a pair of intersecting lines, a line, or even just a point. 3 MAT 109 Precalculus Mathematics 1 11.2 The Parabola Notes L. Sterling September 20th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Parabola A collection of all of the given points, which is P, in the plane that are the same distance, which is d, from the ▯xed point, which is F, as they are based on the ▯xed line, which is D. 1.1 Parabola Formula d(F; P) = d(P; D) d(Focus; Point) = d(Point; Directrix) 2 Focus The point, which is F, of a parabola. 1 3 Directrix The line, which is D, of a parabola. 4 Axis of Symmetry When the line, which is not meaning the directrix, is going through F, which is the focus and is perpendicular to D, which is the directrix. 5 Vertex A parabolas given point of intersection, which is V , is with its axis of symmetry. 5.1 Vertexs Satisfying Equation d(F; V ) = d(V; D) d(Focus; V ertex) = d(V ertex; Directrix) 6 Equation of a Parabola q 2 2 (x ▯ a) + (y ▯ 0) = jx + aj q 2 (x ▯ a) + y = jx + aj (x ▯ a) + y = (x + a) 2 2 2 2 2 x ▯ 2ax + a + y = x + 2ax + a ▯ ▯ y = x + 2ax + a ▯ x ▯ 2ax + a 2 2 y = 4ax 2 2 ▯ y = 4ax { Vertex: (0;0) { Focus: (a;0) { : Directrix: x = ▯a; a > 0 7 Latus Rectum The given line segment is joining two points, which can be (a;2a) and (a;▯2a), with 4a being its length. 8 Parabola Equations: Vertex at (0;0) 8.1 Vertex of (0;0), a > 0, and Focus of (a;0) ▯ Directrix: x = ▯a 2 ▯ Equation: y = 4ax ▯ Axis of Symmetry: X-axis ▯ Opens: Right 8.2 Vertex of (0;0), a > 0, and Focus of (▯a;0) ▯ Directrix: x = a 2 ▯ Equation: y = ▯4ax ▯ Axis of Symmetry: X-axis ▯ Opens: Left 3 8.3 Vertex of (0;0), a > 0, and Focus of (0;a) ▯ Directrix: y = ▯a 2 ▯ Equation: x = 4ay ▯ Axis of Symmetry: Y -axis ▯ Opens: Up 8.4 Vertex of (0;0), a > 0, and Focus of (0;▯a) ▯ Directrix: y = a ▯ Equation: x = 4ay ▯ Axis of Symmetry: Y -axis ▯ Opens: Down 9 Parabola Equations: Vertex at (h;k) 9.1 Vertex of (h;k), a > 0, and Focus of (h + a;k) ▯ Directrix: x = h ▯ a 2 ▯ Equation: (y ▯ k) = 4a(x ▯ h) ▯ Axis of Symmetry: Parallel to the X-axis ▯ Opens: Right 9.2 Vertex of (h;k), a > 0, and Focus of (h ▯ a;k) ▯ Directrix: x = h + a ▯ Equation: (y ▯ k) = ▯4a(x ▯ h) 4 ▯ Axis of Symmetry: Parallel to the X-axis ▯ Opens: Left 9.3 Vertex of (h;k), a > 0, and Focus of (h;k ▯ a) ▯ Directrix: y = k ▯ a ▯ Equation: (x ▯ h) = 4a(y ▯ k) ▯ Axis of Symmetry: Parallel to the Y -axis ▯ Opens: Up 9.4 Vertex of (h;k), a > 0, and Focus of (h;k + a) ▯ Directrix: y = k + a ▯ Equation: (x ▯ h) = ▯4a(y ▯ k) ▯ Axis of Symmetry: Parallel to the Y -axis ▯ Opens: Down 10 Paraboloid of Revolution A given surface that is formed by rotating a parabola about its own axis of symmetry. 5 MAT 109 Precalculus Mathematics 1 11.3 The Ellipse Notes L. Sterling September 21st, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Ellipse A general collection of all points in a plane. 2 Foci A constant that is also the sum of the distances from the two ▯xed points. 3 Major Axis The given line that contains the foci. 1 4 Center The general line segments midpoint that is joining the foci. 5 Minor Axis The given line that goes through the center and is also perpendicular to the major axis. 6 Vertices The two points of intersection from both the ellipse and major axis. 7 Major Axis Length The distance from one vertex to another vertex. 8 Ellipse’s Equation: Major Axis along the X- Axis 2 2 x + y = 1 a2 b a > b > 0 b = a ▯ c2 Center : (0; 0) Foci : (▯c; 0) and (c; 0) V ertices : (▯a; 0) and (a; 0) 2 Major Axis : X ▯ Axis 9 Ellipse’s Equation: Major Axis along the Y - Axis x2 y2 b2+ a2= 1 a > b > 0 b = a ▯ c2 Center : (0; 0) Foci : (0; ▯c) and (0; c) V ertices : (0; ▯a) and (0; a) Major Axis : Y ▯ Axis 10 Ellipse’s Equation: Center at (h;k) Major Axis Parallel to a Coordinate Axis 10.1 Center of (h;k) with Major Axis parallel to the X- Axis Foci : (h ▯ c; k) and (h + c; k) V ertices : (h ▯ a; k) and (h + a; k) 2 2 (x ▯ h) (y ▯ h) Equation : a2 + b = 1 3 a > b > 0 b = a ▯ c 2 10.2 Center of (h;k) with Major Axis parallel to the Y - Axis Foci : (h; k ▯ c) and (h; k + c) V ertices : (h; k ▯ a) and (h; k ▯ a) 2 2 (x ▯ h) (y ▯ h) Equation : b2 + a 2 = 1 a > b > 0 2 2 2 b = a ▯ c 4 MAT 109 Precalculus Mathematics 1 11.4 The Hyperbola Notes L. Sterling September 22nd, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Hyperbola The general collection of all of the points in the plane. 2 Foci A constant that is also the di▯erence of the distances from the two ▯xed points. 3 Transverse Axis The given line that contains the foci. 1 4 Center The general line segments midpoint that is joining the foci. 5 Conjugate Axis The given line that goes through the center and is also perpendicular to the transverse axis. 6 Branches The two separate curves of a hyperbola that are symmetric, but with respect to the traverse axis, center, and the conjugate axis. 7 Vertices The two points of intersection from both the hyperbola and transverse axis. 8 Major Axis Length The distance from one vertex to another vertex. 9 Hyperbola Equation: Center at (0;0) 9.1 Transverse Axis along the X-Axis 2 2 x ▯ y = 1 a2 b2 b = c ▯ a2 Center : (0; 0) 2 Foci : (▯c; 0) and (c; 0) V ertices : (▯a; 0) and (a; 0) Transverse Axis : X ▯ Axis 9.2 Transverse Axis along the Y -Axis 2 2 y ▯ x = 1 a2 b2 2 2 2 b = c ▯ a Center : (0; 0) Foci : (0; ▯c) and (0; c) V ertices : (0; ▯a) and (0; a) Transverse Axis : Y ▯ Axis 10 Asymptotes of a Hyperbola 2 y2 10.1 Hyperbola : x2 ▯ 2 = 1 a b Two Oblique Asymptote ▯ ▯ ▯ ▯ b b y = x and y = ▯ x a a y2 2 10.2 Hyperbola : 2 ▯ x2 = 1 a b Two Oblique Asymptote ▯ ▯ ▯ ▯ a a y = x and y = ▯ x b b 3 11 Hyperbolas Equation: Center at (h;k) Transverse Axis Parallel to a Coordinate Axis 11.1 Center of (h;k) and Parallel to the X-Axis Foci : (h ▯ c; k) V ertices : (h ▯ a; k) 2 2 (x ▯ h) (y ▯ k) Equation : a 2 ▯ b2 = 1 b = c ▯ a 2 b y ▯ k = ▯ a (x ▯ h) 11.2 Center of (h;k) and Parallel to the Y -Axis Foci : (h; k ▯ c) V ertices : (h; k ▯ a) (y ▯ k) (x ▯ h)2 Equation : 2 ▯ 2 = 1 a b b = c ▯ a 2 a y ▯ k = ▯ (x ▯ h) b 4

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