Download 9.3 Hyperbolas

January 14, 2015
9.3 Hyperbolas
Objective: Write the standard form of the equation of a
hyperbola, and analyze and sketch the graphs of hyperbolas.
Hyperbola set of all points (x, y) in a plane, the
difference of whose distances from two
distinct fixed points (foci) is constant.
d1
Center - midpoint between the foci.
d2
branches
Vertices - points at which the line segment
through the foci meets the hyperbola.
Transverse axis - line segment joining
the vertices.
*Note: d2 - d1 = positive constant
http://www.brightstorm.com/math/precalculus/conic-sections/the-hyperbola/#ooid=ZzaWV3NDoSM36UvSLAMq-B
http://www.intmath.com/plane-analytic-geometry/6-hyperbola.php
Standard form
e
ot
ot
e
sy
m
pt
pt
m
sy
A
A
Hyperbola with horizontal transverse
Center: (h, k)
Transverse axis of length 2a
Conjugate axis of length 2b
Vertices: (h ± a, k)
(x-h)2 (y-k)2
=1
Foci: (h ± c, k)
a2
b2
Asymptote: y = k± b
a (x - h)
le"
ang
ect
R
gic
"Ma
sy
m
pt
A
pt
m
sy
e
ot
NOTE: In both cases, c2 = a2 + b2.
a = distance from center to the vertices
b = distance from center to endpoint of conjugate axis
c = distance from the center to the foci
A
Transverse axis of length 2a
"
Conjugate axis of length 2b
gle
tan
c
e
Vertices: (h, k ± a)
R
gic
"Ma
Foci: (h, k ± c)
Asymptote: y = k ± a (x - h) (y-k)2 (x-h)2
b
=1
a2
b2
ot
e
Hyperbola with vertical transverse
Center: (h, k)
January 14, 2015
Ex1: Find the center, vertices, asymptotes, and foci of the hyperbola
given by 4y2 - 9x2 = 36.
a = 3, b = 2, and c2 = 9 + 4 ⇒ c ≈ 3.6. So, the center is (0, 0), the
vertices are at (0, ±3), the asymptotes are y = ±(3/2)x, and the foci
are at (0, ±3.6).
Ex2: Find the standard form of the equation of the hyperbola
centered at the origin with transverse axis of length 4 and foci at
(±3, 0).
We know that a = 2 and c = 3. Next b2 = 32 - 22 or b =± √5. So, the
equation is x2/4 - y2/5 = 1.
Ex3: Sketch the graph of the hyperbola given by
4x2 - 9y2 - 24x - 72y - 72 = 0.
Eccentricity
The eccentricity of a hyperbola is given by e = ca and e > 1.
the larger the eccentricity, the closer the branches of the
hyperbola are to being lines.
January 14, 2015
Applications
Orbits of comets and planets p.654
http://mathforum.org/~sanders/geometry/GP20Hyperbola.html
Ex4: There is a listening station located at A(2200, 0) (in feet) and
another at B(-2200, 0). An explosion is heard at station A one second
before it is heard at station B. Where was the explosion located?
*Hint: Speed of sound = 1100 ft/sec
The explosion o
c = 2200. Sinc
second, 2a = 1
b = 2200 - 5
right branch of
2
Classifying Conics from its General Equation
The graph of Ax + Cy + Dx + Ey + F = 0 is:
1. a circle if A = C.
2
2
2. a parabola if AC = 0.
3. an ellipse if AC > 0.
4. a hyperbola if AC < 0.
Ex5: Classify each of the following.
a) ellipse (AC = 20)
a) 4x + 5y - 9x + 8y = 0
b) parabola (AC = 0)
b) 2x - 5x + 7y - 8 = 0
c) circle (A = C)
c) 7x + 7y - 9x + 8y - 16 = 0
d) hyperbola (AC = -20)
d) 4x - 5y - x + 8y + 1 = 0
2
2
2
2
2
2
2
2
January 14, 2015