4.4 conics recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and...
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4.4 Conics
• Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas.
• Write the equation and find the focus of a
parabola.
• Write the equation of a ellipse and find the
foci, vertices, the length of the major and minor
axis.
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Parabolas
Definition: A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line, called the directrix, and a fixed point, the focus, not on the line.
Directrix
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Parabolas
x2 = 4py p 0Vertex (0, 0)Directrix y = -pFocus (0, p)Line of sym x = 0
y2 = 4px p 0Vertex (0,0)Directrix x = -pFocus (p, 0)Line of sym y = 0
Standard equation of the Parabola
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Parabola ExamplesGiven 22xy Find the focus. Since the squared variable is x, the
parabola is oriented in the y directions. The leading coefficient is negative, so, the parabola opens down.
-pp
Focus (0, p)
•
Solve for x2
pyyx
yx
42
12
1
2
2
Solve for p.
p
p
8
1
42
1
Focus )8
1,0(
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Parabola ExamplesWrite the standard form of the equation of the parabola with the vertex at the origin and the focus (2, 0).
Note that the focus is along the x axis, so the parabola is oriented in the x axis direction, y2 = 4px.
Focus (2, 0) (p, 0)
p = 2
y2 = 4px = 4(2)x
y2 = 8x
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Classwork
Page 370 problems 9 –14.Page 371 problems 17 – 28.
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Ellipse
Definition: An ellipse is the set of all points (x, y ) in a plane the sum of whose distances from two distinct fixed points (foci) is constant.
• •
•
Focus Focus
(x, y)
d1 d2
d1 + d2 = constant
Vertex Vertex
MajorAxis
MinorAxis
•
•
•
• •Center
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Ellipse
The standard form of the equation of an ellipse (Center at origin)
1or 12
2
2
2
2
2
2
2
a
y
b
x
b
y
a
x
where 0 < b < a c2 = a2 – b2
Major axis length = 2a Minor axis length = 2b
Major axis along the x axis
Major axis along the y axis
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Ellipse Examples
Given 4x2 + y2 = 36. Find the vertices, the end points of the minor axis, the foci and center.
Change the equation to the standard form.
1369
36
36
3636
4
22
2
yx
yx
a2b2
Major axis along the y-axis
Center (0, 0) Vertices (0, 6) End points of minor axis (3, 0)
To find the foci use c2 = a2 – b2
c2 = 36 – 9 c2 = 27
)33,0( Foci
33
c
(0, a) (b, 0)
(0, c)
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Classwork
Page 372 problems 35 – 40 45 – 55
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Hyperbolas
Definition: a hyperbola is the set of all points (x, y) the difference of whose distances from two distinct points (foci) is constant.
••
•
(x, y) ••
•
d1
d2
d1 - d2 = constant
••
•Center
Vertex
Vertex
Focus
Focus
Transversal Axis
Branch
Branch