Conic Sections: Parabolas, Ellipses, and Hyperbolas, Assignments of Calculus

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Conic Sections

MATH 211, Calculus II

J. Robert Buchanan

Department of Mathematics

Summer 2008

Introduction

The conic sections include the parabola , the ellipse , and the hyperbola.

x

y

x

y

x

y

Example

Example Find the equation of the parabola with focus at ( 1 , 2 ) and directrix y = −2.

-2 -1 (^0 1) x 2 3 4

0

1

2

y

f

d

General Formula for Parabola

Theorem

The parabola with vertex at ( b , c ) , focus at ( b , c +

4 a

) , and

directrix given by y = c −

4 a

has equation y = a ( x − b )^2 + c.

Proof.

Remark: if we switch the roles of x and y we have the following result. Theorem

The parabola with vertex at ( c , b ) , focus at ( c +

4 a

, b ) , and

directrix given by x = c −

4 a

has equation x = a ( y − b )^2 + c.

Example

Example Given the equation for a parabola y = − 2 ( x + 2 )^2 + 1, find the vertex, focus, and directrix.

-5 -4 -3 -2 x -1 0 1

-12.

-7.

-2.

0

y

Reflective Property of Parabolas

If a parabola is thought of as a reflector (for example in a flashlight or satellite dish), all rays traveling perpendicular to the directrix and striking the parabola are reflected through the focus.

d f

Example

Example Suppose an ellipse has foci located at ( 1 , 2 ) and ( 1 , 4 ) and the point with coordinates ( 1 , 1 ) lies on the ellipse. Find the equation of the ellipse.

(^1) -1 (^0 1) x 2 3

2

3

4

5

y

General Formula for Ellipse

Theorem

The equation

( x − x 0 )^2 a^2

( y − y 0 )^2 b^2

= 1 with a > b > 0 describes an ellipse with foci at ( x 0 − c , y 0 ) and ( x 0 + c , y 0 ) where c =

a^2 − b^2_. The_ center of the ellipse is at the point ( x 0 , y 0 ) and the vertices are located at ( x 0 ± a , y 0 ) on the major axis. The endpoints of the minor axis are at ( x 0 , y 0 ± b ).

Remark: the roles of the major and minor axes are reversed when b > a > 0.

Example

Example Identify the following features of the ellipse ( x + 1 )^2 9

( y − 3 )^2 4

(^1) Center (^2) Foci (^3) Vertices (^4) Endpoints of minor axis

Reflective Property of Ellipses

A ray emanating from one focus will always reflect off the ellipse and pass through the other focus.

f f

Example

Example Suppose a hyperbola has foci located at (− 2 , 2 ) and ( 6 , 2 ) and the point with coordinates ( 0 , 2 ) lies on the hyperbola. Find the equation of the hyperbola.

-2 (^0 2) x 4 6

0

2

4

6

y

General Formula for Hyperbola

Theorem

The equation

( x − x 0 )^2 a^2

( y − y 0 )^2 b^2

= 1 describes a hyperbola with foci at ( x 0 − c , y 0 ) and ( x 0 + c , y 0 ) where c =

a^2 + b^2_. The_ center of the hyperbola is at the point ( x 0 , y 0 ) and the vertices are located at ( x 0 ± a , y 0 ). The asymptotes are the lines y = ±

b a

( x − x 0 ) + y 0_._

Theorem

The equation

( y − y 0 )^2 a^2

( x − x 0 )^2 b^2

= 1 describes a hyperbola

with foci at ( x 0 , y 0 − c ) and ( x 0 , y 0 + c ) where c =

a^2 + b^2_. The_ center of the hyperbola is at the point ( x 0 , y 0 ) and the vertices are located at ( x 0 , y 0 ± a ). The asymptotes are the lines y = ±

a b

( x − x 0 ) + y 0_._

Example

Example Identify the following features of the hyperbola x^2 4

( y − 1 )^2 16

(^1) Center (^2) Foci (^3) Vertices (^4) Asymptotes

Reflective Property of Hyperbolas

A ray directed toward one focus will reflect off the hyperbola and travel toward the other focus.

f f