Conic Sections Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2...

Conic Sections

Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

Conic sections are plane figures formed by the intersection of a double-napped cone and a plane.

The conic sections may be defined as sets of points in the plane that satisfy certain geometrical properties.

Parabola Ellipse Hyperbola


A parabola is the set of all points in the plane equidistant from a fixed line and a fixed point not on the line.

The axis is the line passing through the focus and

perpendicular to the directrix.

The vertex is the midpoint of the line segment along the axis joining the directrix to the focus.

parabolaaxis

vertex

focus

directrix

The fixed line is the directrix.The fixed point is the focus.


Note: p is the directed distance from the vertex to the focus.

x

y

y = –p(0, 0)

(0, p)p

x2 = 4py

The standard form for the equation of a parabola with vertex at the origin and a vertical axis is:

x2 = 4py where p 0

focus: (0, p)vertical axis: x = 0 directrix: y = –p,


The standard form for the equation of a parabola with vertex at the origin and a horizontal axis is:

Note: p is the directed distance from the vertex to the focus.

x

y

p (p, 0)

x = –p

(0, 0)

horizontal axis: y = 0, directrix: x = –p

y2 = 4px where p 0

focus: ( p, 0)

y2 = 4px


Example: Find the directrix, focus, and vertex, and sketch the

parabola with equation .

Rewrite the equation in standard form x2 = 4py. x2 = – 8y x2 = 4(–2)y p = –2

vertex: (0, 0)

vertical axis: x = 0

directrix: y = – p y = 2

focus: = (0, p) (0, –2)

2

8

1xy

x

y

y = 2

x = 0

2

8

1xy

(0, –2)

(0, 0)


x

y

Example: Write the standard form of the equation of the parabola with focus (1, 0) and directrix x = –1.

Use the standard from for the equation of a parabola with a horizontal axis: y2 = 4px. p = 1 y2 = 4(1)x. The equation is y2 = 4x.

p = 1

x = -1

(1, 0)

(0, 0)vertex


An ellipse is the set of all points in the plane for which the sum of the distances to two fixed points (called foci) is a positive constant.

The major axis is the line segment passing through the foci with endpoints (called vertices) on the ellipse.

The minor axis is the line segment perpendicular to the major axis passing through the center of the ellipse with endpoints on the ellipse.

major axis

minor axis

center vertexvertex

The midpoint of the major axis is the center of the ellipse.

Ellipsed1 + d2 = constant

d1 d2focus focus


The standard form for the equation of an ellipse with center at

the origin and a major axis that is horizontal is: , with:

ac x

y

ba

(0, 0)(–c, 0) (c, 0)

(0, – b)

(a, 0)

(0, b)

(– a, 0)

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

2 2

2 21

x y

a b

2 2

2 21

x y

a b


x

y

b(0, 0)

ba

The standard form for the equation of an ellipse with center at

the origin and a major axis that is vertical is: , with:

ca

(0, -c)

(0, c)

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

2 2

2 21

x y

b a

(0, – a)

(b, 0)(– b, 0)

(0, a)2 2

2 21

x y

b a


5

4

3

Example: Sketch the ellipse with equation 25x2 + 16y2 = 400 and find the vertices and foci.

1. Put the equation into standard form. 4001625 22 yx divide by 400

25161

400

16

400

25 2222 yx

yx 1

2. Since the denominator of the y2-term

is larger, the major axis is vertical. 3. Vertices: (0, –5), (0, 5)

4. The minor axis is horizontal and intersects the ellipse at (–4, 0) and (4, 0).

5. Foci: c2 = a2 – b2 (5)2 – (4)2 = 9 c = 3 foci: (0, –3), (0,3)

x

y

(4, 0)

(–4, 0)

(0, –3)

(0, 3)(0, 5)

(0, –5)

So, a = 5 and b = 4.


A hyperbola is the set of all points in the plane for which the difference from two fixed points (the foci) is a positive constant.

The graph of the hyperbola has two branches.

The line segment joining the vertices is the transverse axis. Its midpoint is the center of the hyperbola.

transverse axis

d2 – d1 = constant

vertexvertex

center

d1 d2

focusfocus

hyperbola

The line through the foci intersects the hyperbola at two points called vertices.


x

y

The standard form for the equation of a hyperbola with a

horizontal transverse axis is: with:

(c, 0)(–c, 0)focusfocus

(0, b)

(0, –b)

xa

by

asymptote

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

2 2

2 21

x y

a b

vertex (a, 0)

vertex (– a, 0)

A hyperbola with a horizontal transverse axis has asymptotes

with equations and .xa

by x

a

by

asymptote

xa

by


x

y

The standard form for the equation of a hyperbola with a

vertical transverse axis is: with:

(b, 0)(–b, 0)

xb

ay

asymptote

xb

ay

asymptote

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

12

2

2

2

b

x

a

y

A hyperbola with a vertical transverse axis has asymptotes

with equations and .xb

ay x

b

ay

vertex (0, – a)

vertex (0, a)

focus (0, -c)

focus (0, c)


Example: Sketch the hyperbola with equation x2 – 9y2 = 9 and find the vertices, foci, and asymptotes. 1. To write the equation in standard form, divide by 9.

2. Because the x2-term is positive, the transverse axis is horizontal.

3. Vertices: (0, –3), (0, 3)

4. Asymptotes:1

3y x

222 acb 5.

0) ,10( ),0 ,10(foci:

x

y(0, 1)

(0, -1) (3, 0)(-3, 0)

)0 ,10(

xy3

1

xy3

1

.1 ,3113 2

2

2

2

bayx

10)3()1( 222 cc

a = 3 and b = 1

0 ,10

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