Math 160 – Prof. Beydler
Notes Packet #25
Conic Sections (Parabolas, Ellipses, Hyperbolas)
(covers parts of Sullivan 10.1, 10.2, 10.3, and 10.4)
A parabola is defined as the set of all points in a plane equidistant
from a given fixed point (called the focus) and a given fixed line (called
the directrix).
𝑥 2 = 4𝑝𝑦 has focus (0, 𝑝) and directrix 𝑦 = −𝑝.
(if 𝑝 > 0 opens up, if 𝑝 < 0 opens down)
𝑦 2 = 4𝑝𝑥 has focus (𝑝, 0) and directrix 𝑥 = −𝑝.
(if 𝑝 > 0 opens right, if 𝑝 < 0 opens left)
The focal diameter is the distance across the parabola through the focus. It is: |4𝑝|
Ex 1.
Find the focus, directrix, and focal diameter of the parabola, and sketch its graph.
𝑦 2 = −2𝑥
Ex 2.
Find the focus, directrix, and focal diameter of the parabola, and sketch its graph.
𝑥2 + 𝑦 = 0
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Notes Packet #25
Math 160
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An ellipse is defined as the set of all points in a plane whose distances
from two fixed points (called foci) in the plane have a constant sum.
The equation for an ellipse centered at the origin is:
𝑥2
𝑎2
𝑦2
+ 𝑏2 = 1
The 𝑥-intercepts are 𝑥 = ±𝑎, and the 𝑦-intercepts are 𝑦 = ±𝑏.
The foci have a distance of 𝑐 = √𝑎2 − 𝑏2 from the center.
(or if 𝑏 > 𝑎, then 𝑐 = √𝑏2 − 𝑎2 )
The vertices are the points at the ends of the major axis.
Ex 3.
Find the vertices and foci of the ellipse, determine the lengths of the major and minor axes, and
sketch its graph.
25𝑥 2 + 16𝑦 2 = 400
The eccentricity of an ellipse measures how
much the ellipse deviates from a circle. It’s the
ratio of the focal length and the longer semiaxis.
𝑐
Eccentricity =
÷
𝑐
If 𝑎 > 𝑏, then 𝑒 = 𝑎. If 𝑏 > 𝑎, then 𝑒 = 𝑏.
(Note: For all ellipses, 0 < 𝑒 < 1. If more circular, 𝑒 is closer to 0. If more elongated, 𝑒 is closer to 1.)
Ex 4.
Find the eccentricity of the ellipse:
𝑥2
9
+
𝑦2
4
=1
Notes Packet #25
Math 160
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A hyperbola is the set of points in a plane whose distances from two fixed points (called foci) in the
plane have a constant difference.
The equation for a hyperbola centered at the origin
opening horizontally is:
𝑥2
𝑎2
𝑦2
− 𝑏2 = 1
The 𝑥-intercepts are 𝑥 = ±𝑎.
The foci have a distance of 𝑐 = √𝑎2 + 𝑏2 from the center.
𝑏
The hyperbola has asymptotes 𝑦 = ± 𝑎 𝑥.
The vertices are (±𝑎, 0).
Ex 5.
Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph.
𝑥 2 − 3𝑦 2 = 3
The equation for a hyperbola centered at the origin opening vertically is:
𝑦2
𝑏2
𝑥2
− 𝑎2 = 1
The 𝑦-intercepts are 𝑦 = ±𝑏.
The foci have a distance of 𝑐 = √𝑎2 + 𝑏2 from the center.
𝑏
The hyperbola has asymptotes 𝑦 = ± 𝑎 𝑥.
The vertices are (0, ±𝑏).
Notes Packet #25
Math 160
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Note: All conics can be shifted to have center (ℎ, 𝑘 ) by replacing 𝑥 with 𝑥 − ℎ, and 𝑦 with 𝑦 − 𝑘.
ex:
(𝑥−2)2
4
+
(𝑦+1)2
9
= 1 is an ellipse with center (2, −1).
Ex 6.
Determine whether the equation represents a parabola, an ellipse, or a hyperbola.
𝑦 2 − 4𝑥 2 + 2𝑦 + 8𝑥 − 7 = 0
Find the center, foci, vertices, and asymptotes. Sketch the graph.
Math 160
Notes Packet #25
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Ex 7.
Determine whether the equation represents a parabola, an ellipse, or a hyperbola. If the graph is a
parabola, find the vertex, focus, directrix, and focal diameter. If it is an ellipse, find the center, foci,
vertices, and lengths of the major and minor axes. If it is a hyperbola, find the center, foci, vertices,
and asymptotes. Then sketch the graph of the equation.
𝑥 2 + 6𝑥 + 12𝑦 + 9 = 0
Practice for Packet #25
1. Find the vertex, focus, directrix, and focal diameter of the following parabola, and sketch its graph.
1
𝑥 − 2 𝑦2 = 0
Math 160
Notes Packet #25
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2. Find the vertices, foci, and eccentricity of the following ellipse, determine the lengths of the major
and minor axes, and sketch its graph.
9𝑥 2 + 4𝑦 2 = 36
3. Determine whether the equation represents a parabola, an ellipse, or a hyperbola. If the graph is a
parabola, find the vertex, focus, directrix, and focal diameter. If it is an ellipse, find the center,
foci, vertices, lengths of the major and minor axes, and eccentricity. If it is a hyperbola, find the
center, foci, vertices, and asymptotes. Then sketch the graph of the equation.
𝑥 2 − 𝑦 2 = 10𝑥 − 10𝑦 + 1
Math 160
Notes Packet #25
4. Find an equation for the ellipse that satisfies the given conditions.
Foci: (±5, 0), length of major axis: 12
5. Find an equation for the hyperbola that satisfies the given conditions.
1
Vertices: (0, ±6), asymptotes: 𝑦 = ± 𝑥
3
Q: When can you add two to eleven and get one as the correct answer?
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Math 160
Notes Packet #25
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Homework for Packet #25
1. Determine whether the equation represents a parabola, an ellipse, or a hyperbola. If the graph is a
parabola, find the vertex, focus, directrix, and focal diameter. If it is an ellipse, find the center,
foci, vertices, lengths of the major and minor axes, and eccentricity. If it is a hyperbola, find the
center, foci, vertices, and asymptotes. Then sketch the graph of the equation.
1
a) 2 𝑥 2 − 2𝑦 = 0
b) 2𝑥 + 3𝑦 2 = 0
c) 𝑥 2 − 2𝑥 = −6𝑦 − 1
d) 8𝑥 − 12 = 𝑦 2 + 4𝑦
e) 5𝑥 2 + 9𝑦 2 = 45
f) 8𝑥 2 = 4 − 𝑦 2
g) 4𝑥 2 + 𝑦 2 − 8𝑥 + 6𝑦 + 9 = 0
h) 𝑥 2 + 6𝑥 + 4𝑦 2 − 16𝑦 = −9
i) 𝑥 2 = 36𝑦 2 + 9
j) 𝑦 2 − 2𝑥 2 − 4 = 0
k) 9𝑦 2 + 18𝑦 − 𝑥 2 − 4𝑥 = 4
l) 25𝑥 2 − 4𝑦 2 − 150𝑥 + 125 = 0
2. Find an equation for the parabola that satisfies the given conditions.
Focus: (−2, 1), directrix: 𝑥 = 1
3. Find an equation for the parabola that satisfies the given conditions.
Vertex: (4, −2), focus: (6, −2)
4. Find an equation for the ellipse that satisfies the given conditions.
Foci: (0, ±1), vertices: (0, ±2)
5. Find an equation for the ellipse that satisfies the given conditions.
Foci (1, 2) and (−3, 2), vertex: (−4, 2)
6. Find an equation for the hyperbola that satisfies the given conditions.
1
Vertices: (±3, 0), asymptotes: 𝑦 = ± 2 𝑥
7. Find an equation for the hyperbola that satisfies the given conditions.
Center: (−2, 2), vertex: (−2, 5), focus: (−2, 2 + √13)
Optional exercises from the Sullivan book if you’d like more practice:
10.2 (p.646) #13-55 odd
10.3 (p.656) #13-63 odd
10.4 (p.669) #15-35 odd, 41-61 odd