Sect' 8'5 Hyperbolas

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Sect. 8.5 Hyperbolas
Goal 1 Write equations of Hyperbolas Goal 2
Graph Hyperbolas
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• Hyperbola set of all points such that the
  difference of the distances from any point to the
  foci is constant.

Difference of the distances d2 d1
constant
foci
vertices
asymptotes
The Transverse Axis is the line segment joining
the vertices (through the foci)
The midpoint of the transverse axis is the center
of the hyperbola..
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Hyperbolas

• Like an ellipse but instead of the sum of
  distances it is the difference

• The line thru the foci intersects the hyperbola
  the vertices

• The line segment joining the vertices is the
  Transverse Axis, and its midpoint is the Center
  of the hyperbola.

• The Line segment joining the Co-vertices is the
  Conjugate Axis.

• Has 2 branches and 2 asymptotes

• The asymptotes contain the diagonals of a
  rectangle centered at the hyperbolas center

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Asymptotes
(0,b)
Vertex (a, 0)
Vertex (- a, 0)
Focus
Focus
(0,-b)
This is an example of a horizontal Transverse
Axis (a, the biggest number, is under the x2 term
with the minus before the y)
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Standard Equation of a Hyperbola (Center at
Origin)
This is the equation if the transverse axis is
horizontal.
(0, b)
(c, 0)
(c, 0)
(a, 0)
(a, 0)
(0, b)
The foci of the hyperbola lie on the major axis,
c units from the center, where c2 a2 b2
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The Hyperbola
A1 and A2 are the vertices.
Line segment A1A2 is the transverse axis and has
a length of 2a units.
The distance from the center to either focus is
represented by c.
Transverse axis
c
Both the Transverse Axis and Conjugate Axis are
lines of symmetry of the hyperbola.
c
a
a
O
F1
F2
A1
A2
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The Hyperbola Centered at the Origin

• The diagram shows a hyperbola with a rectangle
  centered 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.
• 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
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The Standard Equation of a Hyperbola With
Center (0, 0) and Foci on the x-axis

• Length of Transverse axis is 2a.
• Length of 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

B (0, b)
(-c, 0)
(c, 0)
A2
A1
(a, 0)
F1
(-a, 0)
F2
B (0, -b)

• The equations of the asymptotes

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The Standard Equation of a Hyperbola with Centre
(0, 0) and Foci on the y-axis
F1(0, c)

• Length of transverse axis is 2a.
• Length of 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)

• The equations of the asymptotes

F2(0, -c)
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Standard Equation of a Hyperbola (Center at
Origin)
This is the equation if the transverse axis is
vertical.
(0, c)
(0, a)
(b, 0)
(b, 0)
(0, a)
(0, c)
The foci of the hyperbola lie on the major axis,
c units from the center, where c2 a2 b2
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Vertical Transverse Axis
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Standard Form of Hyperbola Center at Origin
Foci lie on transverse axis, c units from the
center c2 a2 b2
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How do you graph a hyperbola?
You need to know center, the vertices, the
co-vertices, and asymptotes.

• The asymptotes intersect at the center of the
  hyperbola and pass through the corners of a
  rectangle with corners ( a, b)

• Example Graph the hyperbola

a 4 b 3
c 5
Draw a rectangle using a and b as the sides...
Draw the asymptotes (diagonals of rectangle)...
(0, 3)
Draw the hyperbola...
(5, 0)
(5,0)
(4,0)
(4, 0)
Here are the equations of the asymptotes Horizon
tal Transverse Axis Vertical Transverse Axis

(0,-3)
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Graph 4x2 9y2 36

• Write in standard form (divide through by 36)

• a 3 and b 2, because the x2 term is
  positive transverse axis is horizontal
  vertices are (-3, 0) (3, 0)

• Draw a rectangle centered at the origin.

• Draw asymptotes.

• Draw hyperbola.

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Write the equation of a hyperbola
with foci (0, - 3) (0, 3) and vertices (0, - 2)
(0, 2).

• Vertical because foci vertices lie on the
  y-axis
• Center at origin because f v are equidistant
  from the origin
• Since c 3 a 2, c2 b2 a2
• 9 b2 4
• 5 b2

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Write the equation 4x2 16y2 64
in standard form. Find the foci and vertices
of the hyperbola.
Get the equation in standard form (make it equal
to 1)
Simplify...
That means a 4 b 2
Use c2 a2 b2 to find c. c2 42
22 c2 16 4 20 c
(0, 2)
(4,0)
(4, 0)
(c, 0)
(c, 0)
Vertices Foci
(0,-2)
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Write an equation of the
hyperbola whose foci are (0, 6)
and (0, 6) and whose vertices are
(0, 4) and (0, 4).
Since the major axis is vertical, the equation is
the following
The center is (0, 0).
(0, 6)
Since a 4 and c 6 , find b... c2
a2 b2 62 42 b2 36 16
b2 20 b2 The equation of the hyperbola
(0, 4)
(b, 0)
(b, 0)
(0, 4)
(0, 6)
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Analyze the 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
the equation.
The equations of the asymptotes are
For this equation, a 2 and b 4. Length of
transverse axis is 2a 4. Length of 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
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Analyze the Hyperbola
For this equation, a 5 and b 3. Length of
transverse axis is 2a 10. Length of 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
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Standard Form of the Hyperbola with Center (h, k)
When the transverse axis is vertical, the
equation in standard form is
The center 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)
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Standard Form of the Hyperbola with Center (h, k)
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
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Find the Equation of a Hyperbola
The center is (2, 3), so h 3 and k
2. Transverse axis is parallel to the y-axis and
has a length of 10 units, so a 5. 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
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Write the Equation of the Hyperbola
Write the equation of the hyperbola with center
at (2, -3), one vertex at (6, -3), and the
coordinates of one focus at (-3, -3).
The center is (2, -3), so h 2, k -3.
The distance from the center to the vertex is 4
units, so a 4. The distance from the center to
the foci is 5 units, so c 5.
b2 c2 - a2 25 - 16 9 b 3
Standard form
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Analyze the 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 32x ) (- 9y2
18y) 91 0 4(x2 8x ____) - 9(y2 - 2y
_____) -91 _____ _____
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1
64
-9
4(x 4)2 - 9(y - 1)2 -36
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The center is (-4, 1).
For this equation, a 2 and b 3. Length of
transverse axis is 2a 4. Length of 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
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Graph the hyperbola defined by 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
The center is (2, -1).
c2 a2 b2 6 4 10
Foci
The equations of the asymptotes are
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SOLUTION
The y 2-term is positive, so the transverse axis
is vertical. Since a 2 1 and b 2 4, you know
that a 1 and b 2.
Plot the center at (h, k) (1, 1). Plot the
vertices 1 unit above and below the center at
(1, 0) and (1, 2).
Draw a rectangle that is centered at (1, 1) and
is 2a 2 units high and 2b 4 units
wide.
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SOLUTION
The y 2-term is positive, so thetransverse axis
is vertical. Sincea 2 1 and b 2 4, you know
thata 1 and b 2.
Draw the asymptotes through the corners of the
rectangle.
Draw the hyperbola so that it passes through the
vertices and approaches the asymptotes.
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Graph the hyperbola 9x2 4y2 18x 16y 43
0
Group the xs and ys together...
Factor out the GCFs...
9x2 18x 4y2 16y 43
Complete the squares ...
9(x2 2x ) 4(y2 4y ) 43
1 4
9 16
9(x 1)2 4(y 2)2 36
9(x 1)2 4(y 2)2 36 36 36
36
Center (1, 2)
a 2 b 3