Hyperbola PowerPoint

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Chapter 3 Conics
3.5
The Hyperbola
3.5.1
MATHPOWERTM 12, WESTERN EDITION
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The Hyperbola
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.
A1 and A2 are called the vertices.
Line segment A1A2 is called the transverse axis
and has a length of 2a units.
Transverse axis
The distance from the centre to either focus is
represented by c.
c
c
a
a
O
F1
F2
A1
A2
Both the transverse axis and its perpendicular bis
ector are lines of symmetry of the hyperbola.
3.5.2
3
Locus Definition
P(x, y)
F1
F2
PF1 - PF2 2a
3.5.3
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The Hyperbola Centred at the Origin
The diagram shows a graph of a hyperbola with a
rectangle centred at the origin. The points A1,
A2 , B1 and B2 are the midpoints of the sides of
the rectangles. The hyperbola lies between the
lines containing its diagonals. As x
increases, the hyperbola comes closer to these
lines. These lines are asymptotes. The line
segment B1B2 is called the conjugate axis. The
conjugate axis has a length of 2b units.
B1
A1
A1
B2
For a hyperbola, the value of b can be
found using the Pythagorean Theorem, a2 b2
c2.
3.5.4
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The Standard Equation of a Hyperbola With
Centre (0, 0) and Foci on the x-axis
The equation of a hyperbola with the centre (0,
0) and foci on the x-axis is
B (0, b)
The length of the transverse axis is 2a. The
length of the conjugate axis is 2b. The vertices
are (a, 0) and (-a, 0). The foci are (c, 0) and
(-c, 0). The slopes of the asymptotes are
(-c, 0)
(c, 0)
A2
A1
(a, 0)
F1
(-a, 0)
F2
B (0, -b)
The equations of the asymptotes
3.5.5
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The Standard Equation of a Hyperbola with Centre
(0, 0) and Foci on the y-axis contd
The equation of a hyperbola with the centre (0,
0) and foci on the y-axis is
F1(0, c)
The length of the transverse axis is 2a. The
length of the conjugate axis is 2b. The vertices
are (0, a) and ( 0, -a). The foci are (0, c) and
(0, -c). The slopes of the asymptotes are
A1(0, a)
B2(b, 0)
B1(-b, 0)
A2(0, -a)
F2(0, -c)
The equations of the asymptotes
3.5.6
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Analyzing an 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
each equation.
a)
The equations of the asymptotes are
For this equation, a 2 and b 4. The length of
the transverse axis is 2a 4. The length of the
conjugate axis is 2b 8. The vertices are (2, 0)
and (-2, 0)
c2 a2 b2 4 16 20
The coordinates of the foci are
3.5.7
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Analyzing an Hyperbola
b)
For this equation, a 5 and b 3. The length of
the transverse axis is 2a 10. The length of the
conjugate axis is 2b 6. The vertices are (0, 5)
and (0, -5)
c2 a2 b2 25 9 34
The coordinates of the foci are
The equations of the asymptotes are
3.5.8
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The Standard Form of the Hyperbola with Centre
(h, k)
When the transverse axis is vertical, the
equation in standard form is
The centre is (h, k).
The transverse axis is parallel to the y-axis and
has a length of 2a units. The conjugate axis is
parallel to the x-axis and has a length of 2b
units. The slopes of the asymptotes are
(h, k)
The general form of the equation is Ax2 Cy2
Dx Ey F 0.
3.5.9
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The Standard Form of the Hyperbola with Centre
(h, k) contd
When the transverse axis is horizontal, the
equation in standard form is
The transverse axis is parallel to the x-axis and
has a length of 2a units. The conjugate axis is
parallel to the y-axis and has a length of 2b
units. The slopes of the asymptotes are
3.5.10
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Finding the Equation of a Hyperbola
The centre is (2, 3), so h 3 and k 2. The
transverse axis is parallel to the y-axis and has
a length of 10 units, so a 5. The conjugate
axis is parallel to the x-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 ,
so a 5 and b 3
The coordinates of the foci are
c2 a2 b2 25 9 34
Standard form
3.5.11
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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 where A -25, C 9, D
-150, E -36, F 36.
3.5.12
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Writing the Equation of a Hyperbola
Write the equation of the hyperbola with centre
at (2, -3), one vertex 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
9(x - 2)2 - 16(y 3)2 1
9(x2 - 4x 4) - 16(y2 6y 9) 144 9x2 - 36x
36 - 16y2 - 96y - 144 144 9x2 - 16y2 - 36x -
96y 36 - 144 144 9x2 - 16y2 - 36x -
96y - 216 0
General form
Standard form
3.5.13
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Analyzing an 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
0 4(x2 8x ____) - 9(y2 - 2y _____) -91
_____ _____
16
1
64
-9
4(x 4)2 - 9(y - 1)2 -36
3.5.14
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Analyzing an Hyperbola
The centre is (-4, 1).
For this equation, a 2 and b 3. The length of
the transverse axis is 2a 4. The length of the
conjugate axis is 2b 6. The vertices are (-4,
3) and (-4, -1)
c2 a2 b2 4 9 13
The coordinates of the foci are
The equations of the asymptotes are
3.5.15
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Graphing an Hyperbola
Graph the hyperbola defined by 2x2 - 3y2 - 8x -
6y - 7 0.
2x2 - 3y2 - 8x - 6y - 7
0 (2x2 - 8x) (-3y2 - 6y) -
7 0 2(x2 - 4x ____) - 3(y2 2y ___ ) 7
_____ ______
4
1
8
-3
2(x - 2)2 - 3(y 1)2
12
Standard form
You must enter the equation in the Y editor as y

2(x - 2)2 - 3(y 1)2 12 - 3(y
1)2 12 - 2(x - 2)2
3.5.16
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Graphing the Hyperbola contd
The centre is (2, -1).
c2 a2 b2 6 4 10
The coordinates of the foci are
The equations of the asymptotes are
3.5.17
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General Effects of the Parameters A and C
When A ? C, and A x C lt 0, the resulting conic
is an hyperbola.
When A is positive and C is negative,
the hyperbola opens to the left and right.
When A is negative and C is positive,
the hyperbola opens up and down.
When D E F 0, a degenerate occurs.
9x2 - 4y2 0 (3x - 2y)(3x 2y)
0
E.g., 9x2 - 4y2 0
3x - 2y 0 -2y -3x
or
3x 2y 0 2y -3x
These equations result in intersecting lines.
3.5.18
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Assignment
Pages 159-163 A 1, 3, 6, 8, 11-17 B 19,
20, 23, 25, 27, 33, 36, 50
3.5.19