Transcript 11.4 Hyperbolas - Meglio Education

CONIC
SECTIONS
Part 4: Hyperbola
Hyperbola
Hyperbolas (opening left and right)
(x - h)2 – (y – k)2 = 1
2
a2
b
b
b
Center: (h,k)
Vertices
y= 
(–a, 0) (a, 0)
The Vertices are “a”
distance from the
center.
a
x
y=
a
x
Foci
(–c, 0) (c, 0)
The foci of the hyperbola
lie on the major axis, c
units from the center,
where c2 = a2+ b2
B-points
(–b, 0) (b, 0)
The B-points are “b”
distance from the center.
asymptotes
The asymptotes can be found using
the box method, where the “a” and “b”
points help form a box …..
The
Themajor
transverse
minoraxis
axisis
axis
ishorizontal
vertical
is theOr
line
and
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The
and
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thethe
linear
equations.
reflection.It will
It will
contain
vertices.
the
two “b”
points.
symmetry.
contain
the
Vertices
and
the Foci.
Hyperbolas (opening up and down)
(y - k)2 – (x – h)2 = 1
2
a2
b
a
a
Center: (h,k)
Vertices
y= 
(–a, 0) (a, 0)
The Vertices are “a”
distance from the
center.
b
x
y=
b
x
Foci
(–c, 0) (c, 0)
The foci of the hyperbola
lie on the major axis, c
units from the center,
where c2 = a2+ b2
b-points
(–b, 0) (b, 0)
The b-points are “b”
distance from the center.
asymptotes
The asymptotes can be found using
the box method, where the “a” and “b”
points help form a box …..
The
transverse
axis
is theOr
line
segment
joining
the
Theminor
majoraxis
axisis
ishorizontal
vertical
and
acts
asthe
ofof
you
can
use
linear
equations.
and
acts
asthe
theaxis
line
vertices.
symmetry.
contain
the
Vertices
and
the Foci.
reflection.It will
It will
contain
the
two “b”
points.
Example: Write an equation of the hyperbola with foci (0, –6) and
(0, 6) and vertices (0, –4) and (0, 4). Its center is (0, 0).
vertical
(y – h)2 – (x – k)2 = 1
a2
b2
(0, 6)
(–b, 0)
(0, 4)
(0, –4)
(0, –6)
(b, 0)
a=4, c=6
c2 = a2+ b2
62 = 42 + b2
36 = 16 + b2
20 = b2
The equation of the hyperbola:
y2 – x2 = 1
16
20
Example: Graph
y2 – x2 = 1 ; find foci and asymptotes
9
25
vertical
Draw the rectangle and
a = 3 b = 5 asymptotes...
2
2
2
c =a +b
c2 = 9 + 25 = 34
c = 34
(0, 3)
(–5,0)
(5, 0)
(0,–3)
Foci: 0,  34 and 0, 34
3
Asymptotes: y   x
5
Example:
Write the equation in standard form of 4x2 – 16y2 = 64.
Find the foci and vertices of the hyperbola.
Get the equation in standard form (make it equal to 1):
4x2 – 16y2 = 64
Simplify...
64
64
64
x2 – y2 = 1
16 4
That means a = 4 b = 2
Use c2 = a2 + b2 to find c.
c2 = 42 + 22
c2 = 16 + 4 = 20
(0, 2)
c =  20  2 5
(–4,0)
(4, 0)
(–c,0) (c, 0)
(0,-2)
Vertices: 4,0 and 4,0
 2 5,0 and 2 5,0
Foci:
Example: Graph
(y – 2)2 – (x + 3)2 = 1
25
16
vertical
Center: (–3, 2)
a=5
b=4
To graph, start with the center…
Move 5 units up and down
Move 4 units right and left
Draw the rectangle and asymptotes…
(–3, 7)
(–7, 2)
(1, 2)
(–3, –3)
Example: 9x2 – 4y2 + 18x + 16y – 43 = 0
9x2 + 18x
–4y2 + 16y = 43
9(x2 + 2x + 1) – 4(y2 – 4y + 4 ) = 43 + 9 – 16
9(x + 1)2 – 4(y – 2)2 =36
(x + 1)2 – (y – 2)2 = 1
4
9
c2 = a2+ b2 = 9 + 4
Foci:  1 13, 2
Asymptotes:
y2
3
x  1
2
y2 
3
x  1
2
Center (–1, 2)
a=2 b=3
c  13
Hyperbolas, write equations
Example: Write an equation in standard form for the hyperbola with
vertices (–1, 1) and (7, 1) and foci (–2, 1) and (8, 1).
center: (3, 1)
a= 4
b=
c= 5
c2 = a2 + b2
25 = 16 + b2
b2 = 9
(x – 3)2 – (y – 1)2 = 1
16
9