Chapter 9 Analytic Geometry 2010 Pearson Education Inc

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Chapter 9 Analytic Geometry © 2010 Pearson Education, Inc. All rights reserved 1

Chapter 9 Analytic Geometry © 2010 Pearson Education, Inc. All rights reserved 1

SECTION 9. 1 Conic Sections: Overview OBJECTIVES Most of the sections in this chapter

SECTION 9. 1 Conic Sections: Overview OBJECTIVES Most of the sections in this chapter focus on the plane curves called conics or conic sections. As the name implies, these curves are the sections of a cone (similar to an ice cream cone) formed when a plane intersects the cone. © 2010 Pearson Education, Inc. All rights reserved 2

RIGHT CIRCULAR CONE Draw a circle on a plane. Draw a line l, called

RIGHT CIRCULAR CONE Draw a circle on a plane. Draw a line l, called the axis, passing through the center of the circle and perpendicular to the plane. Choose a point V above the plane on this line. The surface consisting of all lines that simultaneously pass through the point V and the circle is called a right circular cone with vertex V. © 2010 Pearson Education, Inc. All rights reserved 3

RIGHT CIRCULAR CONE The vertex separates the surface into two parts called nappes of

RIGHT CIRCULAR CONE The vertex separates the surface into two parts called nappes of the cone. © 2010 Pearson Education, Inc. All rights reserved 4

Definition s If we slice the cone with a plane, the intersections of the

Definition s If we slice the cone with a plane, the intersections of the slicing plane and the cone are conic sections. If the slicing plane is horizontal (parallel to the first plane), the intersection is a circle. © 2010 Pearson Education, Inc. All rights reserved 5

Definition s If the slicing plane is inclined slightly from the horizontal, the intersection

Definition s If the slicing plane is inclined slightly from the horizontal, the intersection is an ovalshaped curve called an ellipse. © 2010 Pearson Education, Inc. All rights reserved 6

Definition s If the angle of the slicing plane increases so that the slicing

Definition s If the angle of the slicing plane increases so that the slicing plane is parallel to one of the lines generating the cone, the intersection is called a parabola. © 2010 Pearson Education, Inc. All rights reserved 7

Definition s If the angle of the slicing plane increases so that the slicing

Definition s If the angle of the slicing plane increases so that the slicing plane intersects both nappes of the cone, the resulting intersection is called a hyperbola. © 2010 Pearson Education, Inc. All rights reserved 8

Definition s Intersections of the slicing plane through the vertex result in points and

Definition s Intersections of the slicing plane through the vertex result in points and lines called degenerate conic sections. © 2010 Pearson Education, Inc. All rights reserved 9

Definition s The general conic [“section” is implied] is a nonlinear curve or pair

Definition s The general conic [“section” is implied] is a nonlinear curve or pair of chiral curves. This shows the particular linear (or less) cases, namely a point (vertex), a line, and a pair of intersecting lines. © 2010 Pearson Education, Inc. All rights reserved 10

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