Math 1330 Section 8.3
Section 8.3
Hyperbolas
A hyperbola is the set of all points, the difference of whose distances from two fixed points is constant. Each
fixed point is called a focus (plural = foci).
The focal axis is the line passing through the foci.
Basic “Vertical” Hyperbola:
Equation:
y 2 x2
 1
a2 b2
Asymptotes: y  
a
a
x
b
Foci: (0,  c ) , where c 2  a 2  b 2
Vertices: (0,  a )
-a
Eccentricity: e 
c
a
Basic “Horizontal” Hyperbola:
Equation:
x2 y 2

1
a 2 b2
Asymptotes: y  
b
x
a
Foci: (  c, 0) , where c 2  a 2  b 2
-a
a
Vertices: (  a, 0)
Eccentricity: e 
c
a
The transverse axis (length 2a) is the line segment joining the two vertices. The conjugate axis (length 2b) is
the line segment perpendicular to the transverse axis, passing through the center and extending a distance b on
either side of the center.
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Math 1330 Section 8.3
Graphing Hyperbolas:
To graph a hyperbola with center at the origin:

x2 y 2
y 2 x2
Rearrange into the form 2  2  1 or 2  2  1 .
a
b
a b

Decide if it’s a “horizontal” or “vertical” hyperbola.
o if x 2 is positive, it’s horizontal (vertices are on x-axis).
o If y 2 is positive, it’s vertical (vertices are on y-axis).

Use the square root of the number under x to determine how far to measure in x-direction.

Use the square root of the number under y to determine how far to measure in y-direction.

Draw a box with these measurements.

Draw diagonals through the box. These are the asymptotes. Use the dimensions of the box to determine
the slope and write the equations of the asymptotes.

Put the vertices at the edge of the box on the correct axis. Then draw a hyperbola, making sure it
approaches the asymptotes smoothly.

c 2  a 2  b 2 where a 2 and b 2 are the denominators.

The foci are located c units from the center, on the same axis as the vertices.
2
2
To graph a hyperbola with center not at the origin:
 Rearrange (complete the square if necessary) to look like
( x  h) 2 ( y  k ) 2
( y  k ) 2 ( x  h) 2


1
or

 1.
a2
b2
a2
b2
 Start at the center ( h, k ) and then graph it as before.
 To write down the equations of the asymptotes, start with the equations of the asymptotes for the
similar hyperbola with center at the origin. Then replace x with x  h and replace y with y  k .
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Math 1330 Section 8.3
x2 y2

 1 . Find the center, vertices, foci, asymptotes, length and coordinates of both
Example 1: Graph
36
4
Transverse and Conjugate Axes and the eccentricity.
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Math 1330 Section 8.3
Example 2: Graph y 2  25 x 2  8 y  9  0 . Find the center, vertices, foci, asymptotes, length and coordinates
of both Transverse and Conjugate Axes and the eccentricity.
4
Math 1330 Section 8.3
Example 3: Given:
1. Find the center, vertices, foci, asymptotes, length and
coordinates of both Transverse and Conjugate Axes and the eccentricity
5
Math 1330 Section 8.3
Example 4: Use the following information to write the equation for the hyperbola in standard form. Vertices:
(2, 2) and (8, 2), b = 4
Example 5: Use the following information to write the equation for the hyperbola in standard form. Foci are
4, 0 and 4, 0 and the length of conjugate axis is 6.
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Math 1330 Section 8.3
Quiz 15 and 16 help:
1. Determine the angle x in the triangle given below with AB = 8 and BC = 10.
Hint: Use the Law of Sines along with a double-angle formula.
A.
B.
C.
2. In triangle ABC, ∠A measures 33°. If ∠C measures 38° and BC has length 12,
find AB.
°
A.
°
B.
°
°
C. 6
D. sin
°
°
3. Given triangle ABC, the measure of angle A is 30°, the length of BC is 3, and the length of
AC is 5. How many solutions are there for the measure of angle B?
A. 3
B. Can not be determined
C. 0
D. 1
E. 2
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Math 1330 Section 8.3
Math 1330 Popper 18
1. Triangles: ABC and DEC are right triangles with right angles B and E, respectively. If  DE = 6,  EC = 18,  and BE = 12,
find the area of triangle ABC. (Note that the triangle may not be drawn to scale.)
A. 300
B. 150
C. 120
D. 60
2. Find the area of an equilateral triangle with side length 6 feet.
A. 9√3
B. 3√3
C. 12√3
D. 6
3. In triangle ABC, the measure of angle A is 2x, the length of AB is 5, and the length of AC is
√
.
If
sin(x)=
,
what is the area of th81e triangle?
Hint: You will need to use the double angle formula for sine
A. 5
B. 24
C.
D.
√
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Math 1330 Section 8.3
Math 1330 Popper 18
4. Given isosceles triangle ABC with base BC. If each base angle measures 75° and
each leg is 18 inches long, find the area of this triangle.
A. 162
B. 81
C. 81√2
D.
5. ABC is a triangle with AB = 8, BC = 4, and AC = 6. Find cos(A).
Note: You are being asked to find the cosine of A, not the measure of angle A.
Do not use a calculator
A.
B.
C.
D.
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