10.5 Conic Sections 10.6 Conic Sections in Polar Coordinates · One of the most known applications...
Transcript of 10.5 Conic Sections 10.6 Conic Sections in Polar Coordinates · One of the most known applications...
10.5 – Conic Sections
&
10.6 – Conic Sections in Polar Coordinates
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In this section, we will give geometric definitions of
parabolas, ellipses, and hyperbolas and derive their standard
equations.
They are called conic sections or conics because they result
from intersecting a cone with a plane as shown below.
http://www.shodor.org/interactivate/activi
ties/CrossSectionFlyer/
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A parabola is the set of points in a plane that are
equidistant from a fixed point F called the focus and a
fixed line called the directrix.
The point halfway between the focus and the directrix
lies on the parabola and is called the vertex.
The line through the focus perpendicular to the
directrix is called the axis of the parabola.
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An equation of the parabola with focus (0,p) and
directrix y = -p is x2 = 4py.
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An equation of the parabola with focus (p, 0) and
directrix x = -p is y2 = 4px.
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Parabolas with vertex (h,k) and axis parallel to x-axis.
Parabolas with vertex (h,k) and axis parallel to y-axis.
2
2
4 opens right
4 opens left
y k p x h
y k p x h
2
2
4 opens up
4 opens down
x h p y k
x h p y k
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Find an equation for the conic that satisfies
the given conditions.
Parabola, vertical axis,
passing through (1, 5) and vertex (2, 3)
SOLUTION
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Find an equation for the conic that satisfies
the given conditions.
Parabola, horizontal axis,
passing through (-1, 0), (1, -1) and (3, 1)
SOLUTION
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Describe the graph of the equation.
SOLUTION
2 8 6 23 0y x y
One of the most known applications of parabolas is
in describing the path of a projectile thrown in the
air at an angle to the ground.
Other practical application include the design of
automobile headlights, telescopes, and suspension
bridges.
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An ellipse is the set of points in a plane the sum of
whose distances from two fixed points F1 and F2 called
the foci, is a constant.
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The ellipse
has foci (c, 0) where c2 = a2 – b2, and vertices (a, 0).
The line segment connecting the vertices is called the
major axis.
2 2
2 21 0
x ya b
a b
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The ellipse
has foci (0, c) where c2 = a2 – b2,
and vertices (0, a).
The line segment connecting the
vertices is called the major axis.
2 2
2 21 0
x ya b
b a
http://www.shodor.org/interactivate/activities/ConicFlyer/
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Ellipses with center (h,k) and major axis parallel to the x-axis.
Foci: Vertices:
Ellipses with vertex (h,k) and major axis parallel to the y-axis.
Foci: Vertices:
2 2
2 21
x h y kb a
a b
2 2
2 21
x h y kb a
b a
,h c k ,h a k
,h k a ,h k c
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Find an equation for the conic that satisfies
the given conditions.
Ellipse,
foci (0, -1), (8, -1)
vertex (9, -1)
SOLUTION
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Describe the graph of the equation.
SOLUTION
2 216 9 64 54 1 0x y x y
A interesting aspect of ellipses is that if a source of
light or sound is placed at one focus of an surface,
then the light or sound will reflect off the other focus.
A practical application of this aspect of ellipses is
Lithotripsy (a kidney stone treatment).
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A hyperbola is the set of all points in a plane the
difference of whose distances from two fixed points F1
and F2 called the foci, is a constant.
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The hyperbola
has foci (c, 0) where c2 = a2 + b2, vertices (a, 0), and
asymptotes y = (b/a)x.
2 2
2 21
x y
a b
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The hyperbola
has foci (0, c) where c2 = a2 + b2, vertices (0, a), and
asymptotes y = (a/b)x.
2 2
2 21
y x
a b
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Hyperbolas with vertex (h,k) and major axis parallel
to x-axis.
Hyperbolas with vertex (h,k) and major axis parallel
to y-axis.
2 2
2 21
x h y k
a b
2 2
2 21
y k x h
a b
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Find an equation for the conic that satisfies
the given conditions.
Hyperbola,
foci (-3, -7), (-3, 9)
vertices (-3, -4), (-3, 6)
SOLUTION
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Describe the graph of the equation.
SOLUTION
2 23 4 6 8 0 x y x y
Hyperbolas occur frequently as graphs of equations
in Biology, Chemistry, Physics, and Economics.
Boyle’s Law,
Ohm’s Law,
Supply and Demand Curve
Another interesting application was in the navigation
systems developed in WWI and WWII
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Let F be a fixed point called the focus and l be a fixed line called
the directrix in a plane. Let e be a fixed positive number called
the eccentricity. The set of all points P in a plane such that
that is, the ratio of the distance from F
to the distance from l is the constant e
is a conic section. The conic is
a) an ellipse if e < 1
b) a parabola if e = 1
c) a hyperbola if e > 1
| |
| |
PFe
Pl
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A polar equation of the form
represents a conic section with eccentricity e. The
conic is
a) an ellipse if e < 1
b) a parabola if e = 1
c) a hyperbola if e > 1
or 1 cos 1 sin
ed edr r
e e
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Write the polar equation of a conic with the
focus at the origin and the given data.
Ellipse, eccentricity ¾, directrix x =-5
SOLUTION
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Write the polar equation of a conic with the
focus at the origin and the given data.
Ellipse, eccentricity 0.8, vertex (1, /2)
SOLUTION
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Write the polar equation of a conic with the
focus at the origin and the given data.
Hyperbola, eccentricity 3,
directrix r =-6csc
SOLUTION
http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/awl/conics-main.html
http://www.ies.co.jp/math/java/conics/draw_parabola/draw_parabola.html
http://www.ies.co.jp/math/java/conics/focus/focus.html
http://www.ies.co.jp/math/java/conics/sokyok/sokyok.html