Conics
Advanced Math
Section 4.3
Conic
• AKA conic section
• Intersection of a plane and a double-napped
cone
• See figure 4.18 on page 354
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Degenerate conic
• Plane passes through vertex of the cone
• See figure 4.19 on page 354
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Three ways to approach conics
• Intersections of planes and cones
– Original Greeks
• Algebraically
– General second-degree equation
• Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
• Locus (collection) of points satisfying a
general property
– What we’ll use
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Circle
• Section 1.1
• The collection of all points (x, y) that are
equidistant from a fixed point (h, k).
 x  h   y  k 
2
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r
2
5
Parabola
• Set of all points (x, y) in a plane that are
equidistant from a fixed line, the directrix, and a
fixed point, the focus, not on the line. (see figure
4.20 on page 355)
• The vertex is the midpoint between the focus and
the directrix.
• The axis of the parabola is the line passing
through the focus and the vertex.
– Can be vertical or horizontal
– Parabola is symmetric with respect to its axis
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Standard equation of a parabola
• (Vertex at origin) see page 355
directrix y   p
opens up or down
vertical axis
2
x  4 py , p  0
directrix x   p
opens left or right
horizontal axis
2
y  4 px , p  0
• The focus is on the axis p units (directed
distance) from the vertex
• Focus is (0, p) for vertical axis
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• Focus is (p, 0) for horizontal axis
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Examples
• Find the focus and directrix of each
parabola
yx
x  3 y
2
y  6 x
2
1 2
x y
4
2
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Ellipse
• Set of all points (x, y) in a plane the sum of whose
distances from two distinct points (foci) is
constant. (See figure 4.25 on page 357)
• A line through the foci intersects the ellipse at two
vertices.
• The major axis connects the two vertices
• The center is the midpoint of the major axis
• The minor axis is perpendicular to the major axis
at the center
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Standard equation of an ellipse
• (center at origin) see page 357
major axis length 2a
minor axis length 2b
horizontal major axis
0ba
vertical major axis
x2 y 2
 2 1
2
a
b
x2 y 2
 2 1
2
b
a
• Vertices lie on major axis a units from center
• Foci lie on major axis c units from center
c  a b
2
2
2
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Example
• Find the center and vertices of the following
ellipse and sketch its graph
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
5
16 2
x  y  16
9
y
4
2
3
2
1
-5
-4
-3
-2
-1
x
1
2
3
4
5
-1
-2
-3
-4
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11
Hyperbola
• Set of all points (x, y) in a plane the difference of
whose distances from two distinct points (foci) is a
positive constant (see figure 4.30 on page 359)
• Graph has two disconnected branches
• The line through the foci intersects the hyperbola
at two vertices
• The transverse axis connects the vertices
• The center is the midpoint of the transverse axis.
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Standard equation of a hyperbola
• (center at origin) see page 359
horizontal transverse axis
vertical transverse axis
y 2 x2
 2 1
2
a
b
x2 y 2
 2 1
2
a
b
• Vertices lie on transverse axis a units from center
• Foci lie on transverse axis c units from center
c  a b
2
2
2
b 2 Advanced
c 2  Math
a 2 4.3
13
Example
• Find the standard form of the equation of a
hyperbola with center at the origin, vertices
(0, 2) and (0, -2), and foci (0, -3) and (0, 3).
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Asymptotes of a hyperbola
• (center at origin)
horizontal transverse axis
b
y x
a
b
y x
a
vertical transverse axis
a
y x
b
a
y x
b
• Useful for graphing
• Pass through the corners of a rectangle of dimensions 2a
by 2b.
• The conjugate axis has length 2b and joins either (0, b)
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with (0, -b) or (b, 0) with
(-b,
0)
Example
• Sketch the graph of the following hyperbola
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
5
5 y  4 x  20
2
y
4
2
3
2
1
-5
-4
-3
-2
-1
x
1
2
3
4
5
-1
-2
-3
-4
-5
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