conic sections - hyperbolas section 10.4. hyperbola hyperbola – is the set of points (p) in a...
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Conic Sections - Hyperbolas
Section 10.4
Hyperbola
Hyperbola – is the set of points (P) in a plane such that the difference of the distances from P to two fixed points F1 and F2 is a given constant k.
F2F1kPFPF 21
P
Hyperbola
F2F1
Transverse Axis
Vertices = (a, 0)
Asymptotes
xa
by
xa
by
Hyperbola - Equation
For a hyperbola with a horizontal transverse axis, the standard form of the equation is:
F2F11
2
2
2
2
b
y
a
x
P
Hyperbola
F2
F1 Transverse Axis
xb
ay
xb
ay
Hyperbola - Equation
For a hyperbola with a vertical transverse axis, the standard form of the equation is:
F2
F1
12
2
2
2
a
x
b
y
Hyperbola
Definitions:• a – is the distance between the vertex and the center of
the hyperbola• b – is the distance between the tangent to the vertex
and where it intersects the asymptotes• c – is the distance between the foci and the center
Relationships:The distances a, b and c form a right triangle and can be
used to construct the hyperbola.Horizontal_Hyperbola.htmlVertical_Hyperbola.html
Find the Foci
Find the foci for a hyperbola: 1925
22
yx
a2 b2
From the form, we know it’s a horizontal transverse axis. We know the foci are at (c, o ) and that c2 = a2 + b2
34
925
c
Foci are 0,34
Find the Foci
Find the foci for a hyperbola: 12549
22
xy
b2 a2
From the form, we know it’s a vertical transverse axis. We know the foci are at (0, c ) and that c2 = a2 + b2
74
2549
c
Foci are 74,0
Write the Equation
Write the equation of the hyperbola with foci at (5, 0) and vertices at (3, 0)
From the info, it’s a horizontal transversal. We need to find b
c a
b
b
b
b
4
16
925
35
2
2
222
1169
22
yx
Write the Equation
Write the equation of the hyperbola with foci at (0, 13) and vertices at (0, 5)
From the info, it’s a vertical transversal. We need to find a
c b
2
2
222
144
25169
513
a
a
a
114425
22
xy
Assignment