10.5 – Conic Sections

&

10.6 – Conic Sections in Polar Coordinates

2

In this section, we will give geometric definitions of

parabolas, ellipses, and hyperbolas and derive their standard

equations.

They are called conic sections or conics because they result

from intersecting a cone with a plane as shown below.

http://www.shodor.org/interactivate/activi

ties/CrossSectionFlyer/

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A parabola is the set of points in a plane that are

equidistant from a fixed point F called the focus and a

fixed line called the directrix.

The point halfway between the focus and the directrix

lies on the parabola and is called the vertex.

The line through the focus perpendicular to the

directrix is called the axis of the parabola.

4

An equation of the parabola with focus (0,p) and

directrix y = -p is x2 = 4py.

5

An equation of the parabola with focus (p, 0) and

directrix x = -p is y2 = 4px.

6

Parabolas with vertex (h,k) and axis parallel to x-axis.

Parabolas with vertex (h,k) and axis parallel to y-axis.

2

2

4 opens right

4 opens left

y k p x h

y k p x h

2

2

4 opens up

4 opens down

x h p y k

x h p y k

7

Find an equation for the conic that satisfies

the given conditions.

Parabola, vertical axis,

passing through (1, 5) and vertex (2, 3)

SOLUTION

8

Find an equation for the conic that satisfies

the given conditions.

Parabola, horizontal axis,

passing through (-1, 0), (1, -1) and (3, 1)

SOLUTION

9

Describe the graph of the equation.

SOLUTION

2 8 6 23 0y x y

One of the most known applications of parabolas is

in describing the path of a projectile thrown in the

air at an angle to the ground.

Other practical application include the design of

automobile headlights, telescopes, and suspension

bridges.

11

An ellipse is the set of points in a plane the sum of

whose distances from two fixed points F1 and F2 called

the foci, is a constant.

12

The ellipse

has foci (c, 0) where c2 = a2 – b2, and vertices (a, 0).

The line segment connecting the vertices is called the

major axis.

2 2

2 21 0

x ya b

a b

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The ellipse

has foci (0, c) where c2 = a2 – b2,

and vertices (0, a).

The line segment connecting the

vertices is called the major axis.

2 2

2 21 0

x ya b

b a

http://www.shodor.org/interactivate/activities/ConicFlyer/

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Ellipses with center (h,k) and major axis parallel to the x-axis.

Foci: Vertices:

Ellipses with vertex (h,k) and major axis parallel to the y-axis.

Foci: Vertices:

2 2

2 21

x h y kb a

a b

2 2

2 21

x h y kb a

b a

,h c k ,h a k

,h k a ,h k c

15

Find an equation for the conic that satisfies

the given conditions.

Ellipse,

foci (0, -1), (8, -1)

vertex (9, -1)

SOLUTION

16

Describe the graph of the equation.

SOLUTION

2 216 9 64 54 1 0x y x y

A interesting aspect of ellipses is that if a source of

light or sound is placed at one focus of an surface,

then the light or sound will reflect off the other focus.

A practical application of this aspect of ellipses is

Lithotripsy (a kidney stone treatment).

18

A hyperbola is the set of all points in a plane the

difference of whose distances from two fixed points F1

and F2 called the foci, is a constant.

19

The hyperbola

has foci (c, 0) where c2 = a2 + b2, vertices (a, 0), and

asymptotes y = (b/a)x.

2 2

2 21

x y

a b

20

The hyperbola

has foci (0, c) where c2 = a2 + b2, vertices (0, a), and

asymptotes y = (a/b)x.

2 2

2 21

y x

a b

21

Hyperbolas with vertex (h,k) and major axis parallel

to x-axis.

Hyperbolas with vertex (h,k) and major axis parallel

to y-axis.

2 2

2 21

x h y k

a b

2 2

2 21

y k x h

a b

22

Find an equation for the conic that satisfies

the given conditions.

Hyperbola,

foci (-3, -7), (-3, 9)

vertices (-3, -4), (-3, 6)

SOLUTION

23

Describe the graph of the equation.

SOLUTION

2 23 4 6 8 0 x y x y

Hyperbolas occur frequently as graphs of equations

in Biology, Chemistry, Physics, and Economics.

Boyle’s Law,

Ohm’s Law,

Supply and Demand Curve

Another interesting application was in the navigation

systems developed in WWI and WWII

25

Let F be a fixed point called the focus and l be a fixed line called

the directrix in a plane. Let e be a fixed positive number called

the eccentricity. The set of all points P in a plane such that

that is, the ratio of the distance from F

to the distance from l is the constant e

is a conic section. The conic is

a) an ellipse if e < 1

b) a parabola if e = 1

c) a hyperbola if e > 1

| |

| |

PFe

Pl

26

A polar equation of the form

represents a conic section with eccentricity e. The

conic is

a) an ellipse if e < 1

b) a parabola if e = 1

c) a hyperbola if e > 1

or 1 cos 1 sin

ed edr r

e e

27

28

29

Write the polar equation of a conic with the

focus at the origin and the given data.

Ellipse, eccentricity ¾, directrix x =-5

SOLUTION

30

Write the polar equation of a conic with the

focus at the origin and the given data.

Ellipse, eccentricity 0.8, vertex (1, /2)

SOLUTION

31

Write the polar equation of a conic with the

focus at the origin and the given data.

Hyperbola, eccentricity 3,

directrix r =-6csc

SOLUTION

http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/awl/conics-main.html

http://www.ies.co.jp/math/java/conics/draw_parabola/draw_parabola.html

http://www.ies.co.jp/math/java/conics/focus/focus.html

http://www.ies.co.jp/math/java/conics/sokyok/sokyok.html