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Material Type: Notes; Class: Precalculus; Subject: (Mathematics); University: University of Houston; Term: Unknown 1989;
Typology: Study notes
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Math 1330Section 8.3HyperbolasA
hyperbola
is the set of all points in the plane so that for every point on the hyperbola, the
difference of its distances from two fixed points is a positive constant. The fixed points are calledthe
foci
of the hyperbola.
The line that passes through the foci is called the
focal axis
. The
center
of the hyperbola is the point that is
midway between the foci.
x
y
2
Basic Equation of a Hyperbola
Center: (0, 0)Foci: (0, -
c ) and (0,
c ), where
Vertices: (0, -
a ) and (0,
a
Length of Transverse Axis: 2
a
Coordinates of the Endpoints of the ConjugateAxis: (
-b
, 0) and (
b , 0)
Length of Conjugate Axis: 2
b
Eccentricity:Equations of the Asymptotes:
(^22)
2 2
2
2
2
c a
e^
=
Translations To graph shift the graph of the hyperbolarespectively, horizontally |
h | units and vertically |
k | units.
If
h
0, the horizontal shift is to the right. If
h
< 0, the horizontal shift is to the left. If
k
0, the vertical shift is upward. If
k < 0, the vertical shift is downward.
, 1
)
(
)
(^
2
2
2
2
=
โ
โ
โ
b
h x
a
k y
, 1
)
(
)
(^
2
2
2
2
=
โ
โ
โ
b
k y
a
h x
, 1
2 2
2 2
=
โ
x b
y a
(^22)
2 2
Example 1: Write the following equation in standard form for the equation of a hyperbola.Example 2: Write the following equation in standard form for the equation of a hyperbola.
2
2
0
29
16
4
18
9
2
2
= โ + + โ โ
y
y
x
x
Example 3: Given the following equationa. Write the equation in standard form.b. State the coordinates of the center.c. State the coordinates of the vertices, and then state the length of the transverse axis.d. State the coordinates of the endpoints of the conjugate axis, and then state the length of the
conjugate axis. e. State the coordinates of the foci.f. State the equations of the asymptotes.g. State the eccentricity.h. Sketch the graph of the hyperbola which includes the features from (b) โ (f). Label the center C, the vertices
and the foci
Example 4: Given the following equationa. Write the equation in standard form.b. State the coordinates of the center.c. State the coordinates of the vertices, and then state the length of the transverse axis.d. State the coordinates of the endpoints of the conjugate axis, and then state the length of the
conjugate axis. e. State the coordinates of the foci.f. State the equations of the asymptotes.g. State the eccentricity.h. Sketch the graph of the hyperbola which includes the features from (b) โ (f). Label the center C, the vertices
and the foci
2
2
2
1
2
1
2
1
2
1
2
2
Example 7: Use the given features of a hyperbola to write an equation for the hyperbola in standard form.Center: (0, 0)Length of the transverse axis: 20Length of the conjugate axis: 16Vertical Transverse AxisExample 8: Use the given features of a hyperbola to write an equation for the hyperbola in standard form.Center: (2, 1)One focus: (7, 1)One vertex: (-1, 1)Example 9: Identify the type of conic section (parabola, ellipse, circle, or hyperbola) represented by thefollowing equation. Do not complete the square and write the equation in standard form; this question can beanswered by looking at the signs of the quadratic terms.Example 10: Identify the type of conic section (parabola, ellipse, circle, or hyperbola) represented by thefollowing equation. Do not complete the square and write the equation in standard form; this question can beanswered by looking at the signs of the quadratic terms.
0
9
3
4
2
=
โ
y
x
x
2
2
