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Conic Sections
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CalculusDe La Salle University
Parabolas
Ellipses
Hyperbolas
Topics:
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CalculusDe La Salle University
Given a quadratic equation
the graph is either a parabola, an ellipse ora hyperbola. For special cases it can be an
empty set, a point or intersecting lines.
GRAPH OF A QUADRATIC EQUATION
Conic Section
022
FEyDxCyBxyAx
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CalculusDe La Salle University
Aparabola is the set of all points in a plane
equidistant from a fixed point and a fixed
line. The fixed point is called the focus whilethe fixed line is called the directrix.
PARABOLA
Parabolas
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CalculusDe La Salle University
axis
focus
parabola
latus rectum
vertex
directrix
Parabolas
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An equation of the parabola having its focus
at the point(0,p) and having as its directrix
the line y = p is
EQUATION OF A PARABOLA
Parabolas
pyx 42
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CalculusDe La Salle University
An equation of the parabola having its focus
at the point(p,0) and having as its directrix
the line x = p is
EQUATION OF A PARABOLA
Parabolas
pxy 42
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CalculusDe La Salle University
Example:
Sketch the graph of the given parabola.
Indicate its vertex, axis, focus, equation of
directrix, and endpoints of latus rectum.1.
2.
3. Focus at (0,-1/2); directrix:
4. Vertex at origin, opening upward thru (-4,-2)
Parabolas
yx 82
yy 122
012 y
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CalculusDe La Salle University
An equation of the parabola with a vertical
axis having its vertex at the point(h,k) and p
as the directed distance from the vertex tothe focus is
PARABOLA IN STANDARD FORM
Parabolas
)(4)(2 kyphx
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CalculusDe La Salle University
An equation of the parabola with a
horizontal axis having its vertex at the point
(h,k) and p as the directed distance from thevertex to the focus is
PARABOLA IN STANDARD FORM
Parabolas
)(4)(2 hxpky
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CalculusDe La Salle University
Example:
Sketch the graph of the given parabola.
Indicate its vertex, axis, focus, equation of
directrix, and endpoints of latus rectum.1.
2.
3. Focus at (1,-4); directrix: x - axis
4. Vertex at (-3,2), opening rightward thru (-2,3)
Parabolas
62
41
xxy
yyx 6122
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An ellipse is the set of points in a plane the
sum of whose distance from two fixed points
is a constant. Each fixed points is called afocus.
ELLIPSE
Ellipses
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minor axis
major axis
focus
vertices center
Ellipses
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If 2a is the constant sum referred to in the
definition of an ellipse and the foci are at
(c,0) and (-c,0), then the equation of theellipse is
where b2 = a2 c2.
EQUATION OF AN ELLIPSE
Ellipses
12
2
2
2
b
y
a
x
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If an ellipse has its foci at (0,c) and (0,-c),
then the equation of the ellipse is
where b2 = a2 c2.
EQUATION OF AN ELLIPSE
Ellipses
12
2
2
2
b
x
a
y
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Example:
Sketch the graph of the given ellipse. Indicate
its vertices, center, foci and endpoints of the
minor axis.
1.
2.
11625
22
yx
1169144
22
yx
Ellipses
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CalculusDe La Salle University
If the center of an ellipse is at (h,k) and the
distance between the two vertices is 2a,
then the equation of an ellipse is of the form
(horizontal major axis)
(vertical major axis)
ELLIPSE IN STANDARD FORM
Ellipses
1)()(
2
2
2
2
b
ky
a
hx
1)()(
2
2
2
2
b
hx
a
ky
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Example:Sketch the graph of the given ellipse. Indicate
its vertices, center, foci and endpoints of the
major and minor axis.1.
2.
3.
4.
01119128150162522
yxyx
0361503625922
yxyx
0115542496 22 yxyx
010554249622
yxyx
Ellipses
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CalculusDe La Salle University
Example:Find the equation of an ellipse in general form
having the given properties then sketch its
graph.1. Vertices at (-5/2,0) and (5/2,0) and a focus at
(3/2,0).
2. Foci at (-1,-1) and (-1,7) and a minor axis oflength 8 units.
Ellipses
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A hyperbola is the set of points in a plane,
the absolute value of the difference of
whose distances from two fixed points is aconstant. Each of the fixed points is called a
focus.
HYPERBOLA
Hyperbolas
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Hyperbolas
asymptoteauxiliary rectangle
conjugate axis
transverse axis
center
foci
vertices
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CalculusDe La Salle University
If 2a is the constant referred to in the
definition of an hyperbola and the foci are
at (c,0) and (-c,0), then the equation of thehyperbola is
where b2 = c2 a2.
EQUATION OF A HYPERBOLA
Hyperbolas
12
2
2
2
b
y
a
x
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If a hyperbola has its foci at (0,c) and (0,-c),
then the equation of the hyperbola is
where b2 = c2 a2.
EQUATION OF A HYPERBOLA
Hyperbolas
12
2
2
2
b
x
a
y
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Example:Sketch the graph of the given hyperbola.
Indicate its vertices, center, foci auxiliary
rectangle and equation of the asymptotes.
1.
2.
1916
22
yx
194
22
xy
Hyperbolas
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If the center of a hyperbola is at (h,k) and
the distance between the two vertices is 2a,
then its equation is of the form
(horizontal transverse
axis)(vertical transverse
axis)
HYPERBOLA IN STANDARD FORM
Hyperbolas
1)()(
2
2
2
2
b
ky
a
hx
1)()(
2
2
2
2
b
hx
a
ky
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CalculusDe La Salle University
Example:Sketch the graph of the given hyperbola.
Indicate its vertices, center, foci auxiliary
rectangle and equation of the asymptotes.1.
2.
02916184922
yxyx
05754249622
yxyx
Hyperbolas
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CalculusD L S ll U i it
Hyperbolas
Example:Find the equation of a hyperbola in general
form having the given properties then sketch
its graph.1. Vertices at (-5,-3) and (-5,-1) and the endpoints
of the conjugate axis at (-7,-2) and (-3,-2).
2. Center at (3,-5), a vertex at (7,-5) and a focusat (8,-5)