Hyperbolas

Section 10.4

1st Definiton

A hyperbola is a conic section formed when a plane intersects both cones.

2nd Definition

A hyperbola is the set of all points (x, y) in the plane the difference of whose distances from two distinct fixed point (foci) is a positive constant.

d1

d2

|d2 – d1| = constant

Turn on the N-Spire Calculator.Open the file Hyperbola Construction.

The graph of a hyperbola has two disconnected branches. The line through the two foci intersects the hyperbola at its two vertices. The line segment connecting the vertices is the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola.

The vertices are a units from the center, and the foci are c units from the center.

c

a

General Equation of a Hyperbola

Ax2 + Cy2 + Dx + Ey + F = 0

Either A or C will be negative.

Standard Equation of a Hyperbola

The standard form of the equation of a hyperbola with center (h, k) is

Transverse axis is horizontal

Transverse axis is vertical

c2 = a2 + b2

2 2

2 21

x h y k

a b

2 2

2 21

y k x h

a b

Each hyperbola has two asymptotes that intersect at the center of the hyperbola. The asymptotes pass through the vertices of a rectangle of dimensions 2a by 2b, with its center at (h, k). The line segment of length 2b joining (h, k + b) and (h, k – b) or (h + b, k) and (h – b, k) is the conjugate axis of the hyperbola.

Asymptotes of a Hyperbola

The equations of the asymptotes of a hyperbola are

Transverse axis is horizontal

Transverse axis is vertical

by k x h

a

ay k x h

b

The eccentricity is .c

ea

Is the eccentricity greater or less than 1? Why?

Examples

Write each hyperbola in standard form then find center, vertices, foci, length of the transverse and conjugate axes, eccentricity, and the equations of the asymptotes. Graph the hyperbola.

1. 2 216 2 128 271 x y x y

2 22 __ 16 8 __ 271 __ __ x x y y

2 22 1 16 8 16 271 1 256 x x y y

2 21 16 4 16 x y

2 21 4

116 1

x y

center:

vertices:

transverse axis:

conjugate axis:

foci:

eccentricity:

(1, 4)(-3, 4), (5, 4)

2a = 8

2b = 2

16 1 17 c

1 17,4

17

4e

asymptotes:

14 1 and

41

4 14

y x

y x

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

V1V2

2 22. 9 4 54 8 41 0 y x y x

2 23 1

14 9

y x

center:vertices:transverse axis:conjugate axis:foci:

eccentricity:

(1, -3)

(1, -5), (1, -1)

2a = 42b = 6

1, 3 13

13

2e

asymptotes:

23 1

3 y x

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

Examples

Given the information, write the hyperbola in standard form.

1. center (0, 0); a = 4, b = 2; horizontal transverse axis

2 2

116 4

x y

2. center (-2, 1); a = 5, c = 8; vertical transverse axis

264 25 b2 39b

2 21 2

125 39

y x

3. vertices at (-5, 1) and (-5, 7); conjugate axis length of 12 units

center: (-5, 4)

vertical transverse axis

2a = 6 so a = 3

2b = 12 so b = 6

2 24 5

19 36

y xEND OF 1ST DAY

If we have a hyperbola with a transverse axis distance of a and an eccentricity of e, each directrix of the hyperbola is defined as the line perpendicular to the line containing the transverse axis at a distance from the center of

Therefore each directrix does not intersect the hyperbola.

.a

eWhy is ?

aa

e

Focus-Directrix Property of Hyperbola

This property explains how the directrix relates to a hyperbola. THIS IS THE THIRD DEFINITION OF A HYPERBOLA.

A hyperbola is the set of all points P such that the distance from a point on the hyperbola to the focus F is e times the distance from the same point to the associated directrix.

PF1 = e • AP

P

F1

Directrix Directrix

F2

A

Example

Graph the hyperbola showing the center, vertices, foci, the asymptotes, and each directrix. Find eccentricity and the equations of the directrix.

16y2 − 9x2 = 1442 2

19 16

y x

eccentricity:

directrix:

5

3

ce

a

31.8

5

3

a

ye

x

y

Example

Find the center, vertices, foci, eccentricity, and equations of asymptotes and the directrix.

9x2 − 4y2 − 90x −24y = −153

2 25 3

14 9

x y

center:

vertices:

foci:

eccentricity:

equations of asymptotes:

equation of each directrix:

(5, -3)(7, -3), (3, -3)

5 13, 3

13

2e

33 5

2 y x

45

13x

Example

Given a hyperbola with a focus at (5, 0), an

associated directrix at , and a point on the

hyperbola at (3, 0), find the eccentricity.

 

9

5x

PF = 2

AP = 3 1.81.2

PF = e • AP

2 1.2 e2

1.2e

5

3e

END OF 2nd DAY

9

5x

Day 3

State the general equation of conic sections.

Ax2 + Cy2 + Dx + Ey + F = 0

A conic section is a(n)

1. circle if

2. parabola if

3. ellipse if

4. hyperbola if

A = CAC = 0

AC > 0

AC < 0

ExampleClassify each of the following. Explain your answers.

1. 4x2 + 5y2 − 9x + 8y = 0

ellipse because AC = 20

2. 2x2 − 5x + 7y − 8 = 0

parabola because AC = 0

3. 7x2 + 7y2 − 9x + 8y − 16 = 0

circle because A = C

4. 4x2 − 5y2 − x + 8y + 1 = 0

hyperbola because AC = -20

Example

A passageway in a house is to have straight sides and a semielliptically-arched top. The straight sides are 5 feet tall and the passageway is 7 feet tall at its center and 6 feet wide. Where should the foci be located to make the template for the arch?

5 feet5 feet6 feet

7 feet

2a = 6

a = 3

b = 7 − 5 = 2 2 23 2 c

5c

2.236 feetc

The foci should be about 2.236 feet right and left of the center of the semiellipse.

ExampleThere is a listening station located at A(2200, 0) (in feet) and another at B(-2200, 0). An explosion is heard at station A one second before it is heard at station B. Where was the explosion located? Sound travels at 1100 feet per second.

The listening stations can be considered foci of a hyperbola. So c = 2200 and the center is (0, 0).

The explosion occurred on the right branch of this hyperbola since station B heard it one second after station A.

d2 d1

d2 − d1 = 1100

d2 − d1 = 2a

2a = 1100

a = 550

The difference is equal to 1100 because station A heard the explosion 1 second before station B.

The location of the explosion is the equation of this hyperbola.

2 2 2 c a b

2 2 22200 550 b2 4,537,500b

2 2

1302,500 4,537,500

x y