MATHPOWERTM 12, WESTERN EDITION 3.5.1

3.5Chapter 3 Conics

The hyperbola is the locus of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to two given points in the plane, the foci, is constant.

F1 F2

ac

O A1 A2

A1 and A2 are called the vertices.

Line segment A1A2 iscalled the transverseaxis and has a length of 2a units.

The distance from thecentre to either focusis represented by c.

Both the transverseaxis and itsperpendicularbisector are lines ofsymmetry of thehyperbola.

ac

Transverseaxis

3.5.2

The Hyperbola

F1 F2

P(x, y)

| PF1 - PF2| = 2a3.5.3

Locus Definition

The diagram shows a graph ofa hyperbola with a rectanglecentred at the origin. The pointsA1, A2 , B1 and B2 are themidpoints of the sides of therectangles. The hyperbolalies between the lines containing its diagonals. As| x | increases, the hyperbolacomes closer to these lines.These lines are asymptotes.The line segment B1B2 iscalled the conjugate axis.The conjugate axis has a length of 2b units.

A1 A1

B1

B2

For a hyperbola, the value of b can be foundusing the Pythagorean Theorem, a2 + b2 = c2.

3.5.4

The Hyperbola Centred at the Origin

(c, 0)(-c, 0)

F1 F2

(-a, 0) (a, 0)A1

A2

B (0, b)

B (0, -b)

The Standard Equation of a Hyperbola With Centre (0, 0) and Foci on the x-axis

The equation of a hyperbolawith the centre (0, 0) andfoci on the x-axis is:

x2

a2 y2

b2 1

The length of the transverseaxis is 2a.The length of the conjugateaxis is 2b.The vertices are (a, 0) and(-a, 0).The foci are (c, 0) and (-c, 0).The slopes of the asymptotes are

b

a and

-b

a.

The equations of the asymptotes

are y =

b

ax and y =

-ba

x.3.5.5

The Standard Equation of a Hyperbola with Centre (0, 0) and Foci on the y-axis [cont’d]

F1(0, c)

F2(0, -c)

A1(0, a)

A2(0, -a)

B2(b, 0)B1(-b, 0)

The equation of a hyperbolawith the centre (0, 0) andfoci on the y-axis is:

y2

a2 x2

b2 1

The length of the transverseaxis is 2a.The length of the conjugateaxis is 2b.The vertices are (0, a) and( 0, -a).The foci are (0, c) and (0, -c).The slopes of the asymptotes are

a

b and

-a

b.

The equations of the asymptotes

are y =

a

bx and y =

-ab

x.

3.5.6

State the coordinates of the vertices, the coordinates of the foci, the lengths of the transverse and conjugate axes, and the equations of the asymptotes of the hyperbola defined by each equation.

x2

4

y2

161

a)

For this equation, a = 2 and b = 4.The length of the transverse axis is2a = 4.The length of the conjugate axis is2b = 8.The vertices are (2, 0) and (-2, 0):

c2 = a2 + b2

= 4 + 16 = 20c 20c 2 5

The coordinates of the foci are

(2 5,0) and ( 2 5,0).

The equations of the asymptotes are

y

2

1x and y

2

1x .

3.5.7

Analyzing an Hyperbola

y2

25

x2

91b)

c2 = a2 + b2

= 25 + 9 = 34c 34

The coordinates of the foci are

(0, 34) and (0, 34).

The equations of the asymptotes are

y

5

3x and y

5

3x.

For this equation, a = 5 and b = 3.The length of the transverse axis is2a = 10.The length of the conjugate axis is2b = 6.The vertices are (0, 5) and (0, -5):

3.5.8

Analyzing an Hyperbola

The Standard Form of the Hyperbola with Centre (h, k)

(h, k)

The centre is (h, k).

When the transverse axis is vertical, the equation in standard form is:

(y k)2

a2 (x h)2

b2 1

The transverse axis is parallel to they-axis and has a length of 2a units.The conjugate axis is parallel to thex-axis and has a length of 2b units.The slopes of the asymptotes are

a

b and

-a

b.

The general form of the equation isAx2 + Cy2 + Dx + Ey + F = 0.

3.5.9

When the transverse axis is horizontal, the equation in standard form is:

(x h)2

a2 (y k)2

b2 1

The transverse axis is parallel to thex-axis and has a length of 2a units.The conjugate axis is parallel to they-axis and has a length of 2b units.The slopes of the asymptotes are

b

a and

-b

a.

3.5.10

The Standard Form of the Hyperbola with Centre (h, k) [cont’d]

Finding the Equation of a HyperbolaThe centre is (2, 3), so h = -2 and k = 3.The transverse axis is parallel to they-axis and has a length of 10 units, so a = 5.The conjugate axis is parallel to thex-axis and has a length of 6 units, so b = 3.The vertices are (-2, 8) and (-2, -2).

The slope of one asymptote is ,

(y k)2

a2 (x h)2

b2 1

(y 2)2

52 (x 3)2

32 1

(y 2)2

25

(x 3)2

91

Standardform

c2 = a2 + b2

= 25 + 9 = 34c 34

The coordinates of the foci are

( 2, 3 34 ) and ( 2, 3 34).

3.5.11

5

3so a = 5 and b = 3:

(y 2)2

25

(x 3)2

91

Writing the Equation in General Form

9(y - 2)2 - 25(x - 3)2 = 225 9(y2 - 4y + 4) - 25(x2 - 6x + 9) = 225 9y2 - 36y + 36 - 25x2 - 150x - 225 = 225 -25x2 +9y2 - 150x - 36y + 36 - 225 = 225 -25x2 + 9y2 - 150x - 36y - 414 = 0

The general form of the equation is -25x2 + 9y2 - 150x - 36y + 36 = 0 whereA = -25, C = 9, D = -150, E = -36, F = 36.

3.5.12

Write the equation of the hyperbola with centre at (2, -3), onevertex at (6, -3), and the coordinates of one focus at (-3, -3).

The centre is (2, -3), so h = 2, k = -3.

The distance from the centre to the vertex is 4 units, so a = 4.The distance from the centre to the foci is 5 units, so c = 5.

Use the Pythagorean property to find b:

b2 = c2 - a2

= 25 - 16 = 9 b = ± 3

(x h)2

a2 (y k)2

b2 1

(x 2)2

42 (y 3)2

32 1

(x 2)2

16

(y 3)2

91

9(x - 2)2 - 16(y + 3)2 = 1 9(x2 - 4x + 4) - 16(y2 + 6y + 9) = 1449x2 - 36x + 36 - 16y2 - 96y - 144 = 1449x2 - 16y2 - 36x - 96y + 36 - 144 = 144 9x2 - 16y2 - 36x - 96y - 216 = 0

Standard form

General form

3.5.13

Writing the Equation of a Hyperbola

State the coordinates of the vertices, the coordinates of the foci, the lengths of the transverse and conjugate axes and the equations of the asymptotes of the hyperbola defined by 4x2 - 9y2 + 32x + 18y + 91 = 0.

4x2 - 9y2 + 32x + 18y + 91 = 0 (4x2 + 32x ) + (- 9y2 + 18y) + 91 = 04(x2 + 8x + ____) - 9(y2 - 2y + _____) = -91 + _____ + _____16 1 64 -9

4(x + 4)2 - 9(y - 1)2 = -36 (x 4)2

9

( y 1)2

41

(y 1)2

4

(x 4)2

91

3.5.14

Analyzing an Hyperbola

c2 = a2 + b2

= 4 + 9 = 13c 13

The coordinates of the foci are

( 4, 1 13) and ( 4, 1 13 ).

The equations of the asymptotes are

y

2

3(x +4)+1 and y

2

3(x +4)+1.

For this equation, a = 2 and b = 3.The length of the transverse axis is2a = 4.The length of the conjugate axis is2b = 6.The vertices are (-4, 3) and (-4, -1):

(y 1)2

4

(x 4)2

91

The centre is (-4, 1).

3.5.15

Analyzing an Hyperbola

Graph the hyperbola defined by 2x2 - 3y2 - 8x - 6y - 7 = 0.

2x2 - 3y2 - 8x - 6y - 7 = 0 (2x2 - 8x) + (-3y2 - 6y) - 7 = 02(x2 - 4x + ____) - 3(y2 + 2y + ___ ) = 7 + _____ + ______ 4 1 8 -3

2(x - 2)2 - 3(y + 1)2 = 12

(x 2)2

6

(y 1)2

41

You must enter the equation in the Y= editor as y = :

(x 2)2

6

(y 1)2

41

2(x - 2)2 - 3(y + 1)2 = 12 - 3(y + 1)2 = 12 - 2(x - 2)2

(y 1)2 12 2(x 2)2

3

y 1 12 2(x 2)2

3

y 12 2(x 2)2

3 1

Standardform

3.5.16

Graphing an Hyperbola

y 12 2(x 2)2

3 1

(x 2)2

6

(y 1)2

41

c2 = a2 + b2

= 6 + 4 = 10c 10

The coordinates of the foci are

(2 10, 1) and (2 10, 1).

The equations of the asymptotes are

y

2

6(x 2) 1 and y

2

6(x 2) 1.

The centre is (2, -1).

3.5.17

Graphing the Hyperbola [cont’d]

3.5.18

General Effects of the Parameters A and C

When A ≠ C, and A x C < 0, the resulting conic is an hyperbola.

When A is positive and C is negative, thehyperbola opens to the left and right.

When A is negative and C is positive, thehyperbola opens up and down.

When D = E = F = 0, a degenerate occurs.

E.g., 9x2 - 4y2 = 0 9x2 - 4y2 = 0(3x - 2y)(3x + 2y) = 03x - 2y = 0 -2y = -3x

y 3

2x

or 3x + 2y = 0 2y = -3x

y 3

2x

These equations result in intersecting lines.

Pages 159-163A 1, 3, 6, 8, 11-17B 19, 20, 23, 25, 27, 33, 36, 50

3.5.19