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Conics A conic section is a graph that results from the intersection of a plane and a double cone.
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Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Jan 04, 2016

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Donald Melton
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Page 1: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

ConicsA conic section is a graph that results from the intersection of a plane and a double cone.

Page 2: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

A parabola is the set of all points that are equidistant from a line (the directrix) and a point (the focus). The vertex is the midpoint between the focus and directrix, and the axis is the line through focus and vertex.

Page 3: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

A parabola with vertex (0,0) and directrix y = -p has the equation x2 = 4py

A parabola with vertex (0,0) and directrix x = -p has the equation y2 = 4px

Page 4: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Find the focus of the parabola y = -2x2.

Page 5: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Find the equation of the parabola with vertex at the origin and focus at (2,0).

Page 6: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

An ellipse is the set of all points, the sum of whose distances from two points (foci) is constant. The major axis goes through the foci, the minor axis is perpendicular to the major axis at the center.

Page 7: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

An ellipse centered at (0,0) with horizontal axis length 2a and vertical axis length 2b has equation

The vertices and foci lie on the major (longer) axis. The foci lie c units from the center, where c2 = a2 – b2.

2 2

2 21

x y

a b

Page 8: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Sketch a graph of the ellipse 4x2 + y2 = 36, and identify the vertices and foci.

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Page 9: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Find the equation of the ellipse shown below.

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(2,0)(-2,0)

Page 10: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

A hyperbola is the set of all points, the difference of whose distances from two points (foci) is constant. The transverse axis is the line connecting the vertices, and the midpoint of the transverse axis is the center.

Page 11: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

A hyperbola centered at (0,0) with a horizontal transverse axis has equation

A hyperbola centered at (0,0) with a vertical transverse axis has equation

The foci lie c units from the center, where c2 = a2 + b2.

2 2

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x y

a b

2 2

2 21

y x

b a

Page 12: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Sketch a graph of the hyperbola 4x2 – y2 = 16, and identify the foci.

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Page 13: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Find the equation of the hyperbola shown below.

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(3,0)(-3,0) (-2,0) (2,0)

Page 14: Conics A conic section is a graph that results from the intersection of a plane and a double cone.
Page 15: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Identify and graph

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2 22 1

19 4

x y

Page 16: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Identify and graph

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2 23 2

11 9

x y

Page 17: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Identify and graph (x + 3)2 + (y – 2)2 = 16

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Page 18: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Find the vertex and focus of the parabola x2 – 2x + 4y – 3 = 0

Page 19: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Sketch x2 + 4y2 + 6x – 8y + 9 = 0

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Page 20: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. Sketch -4x2 + y2 + 24x + 4y – 41 = 0

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Page 21: Conics A conic section is a graph that results from the intersection of a plane and a double cone.

Ex. The vertices of an ellipse are (2,-2) and (2,4), and the length of the minor axis is 4. Find the equation of the ellipse.