Conic Sections
©Mathworld
Circle
Circle
• The Standard Form of a circle with a center at (h,k) and a
radius, r, is……..
(x  h)  (y  k)  r
2
2
2
center (3,3)
radius = 2
center (0,0)
radius = 2
Write the standard form of the equation of the circle that goes through the
points:
Plug in
(-10, -5), (-2, 7), (-9,0)
80D12(9D81)  1140
“F” and
solve for
“D”
(10)  (5) 10D  5E  F  0
2
2
(2) 2  (7) 2  2D  7E  F  0
(9) 2  (0) 2  9D  0E  F  0

Plug in “D”
and solve

for “F” and
“E”
10D  5E  F  125
2D  7E  F  53
9D  F  81
Solve for “F”

Combine
first two

equations


Put into
Standard
Form:
F  9D81

70D  35E  7F  875
10D  35E  5F  265
80D12F  1140
D  6

F  9(6) 81 135
10(6) 5E 135 125

x 2  y 2  6x 10 y 135  0
x
2

6x  9 y 2 10 y  25   135  9  25
(x  3)2  (y  5)2  169
Now tell me the center and radius of this circle:
Center: (3, -5)

E 10
Radius: 13
Write the standard form of the equation of the circle that goes through the
points:
(5, 3), (-3, 3), (1, 7)
(x 1)  (y  3)  16
2
Now tell me the center and radius of this circle:
Center: (1, 3)

Radius: 4
2
1.) Write the equation of the circle that passes through the origin and has
its center at (-3, 4).
(x  3)  (y  4)  25
2
2

2.) Write the equation of the circle that passes through the origin and has
its center at (-2, -4).
(x  2)2  (y  4)2  20
Parabolas
What’s in a Parabola
• A parabola is the set of all points in a plane such
that each point in the set is equidistant from a
line called the directrix and a fixed point called
the focus.
Why is the focus so important?
Parabolas
•
2
y

ax
 bx  c
Quadratic Form:
The Standard Form of a Parabola that has a vertex at (h,k) is……
y  a(x  h)  k
2
a > 0 open up
a < 0 open down
Axis of Symmetry:
x=h
Directrix:
y = ???
x  a(y  k)  h
2
a > 0 open right
a < 0 open left

y=k
x = ???
1 distance from vertex and focus, OR distance from vertex to directrix
4a
1
a the length of the latus rectum
x 2  4x  2y 10  0
Graph this equation:

Standard Form:
Vertex:
Focus:
y
1
(x  2)2  3
2
(-2, -3)

(-2, -3.5)
Directrix: y = -2.5
Axis of Symmetry: x = -2
x 2  2x 12y 13  0
Graph this equation:

Standard Form:
Vertex:
(1,
 1)
Focus:
(1, 4)
Directrix:
y
1
(x 1)2 1
12
y = -2
Axis of Symmetry: x = 1
More Problems
1.) Find the focus, vertex and directrix:
3x + 2y2 + 8y – 4 = 0
2
2
x  y  2  4
3
Vertex: (4, -2)

Focus: 4 3 ,  2

Directrix: x  20

3

2.) Find the equation in standard form of a parabola
with directrix x = -1 and focus (3, 2).

1
2
x  y  2 1
8
3.) Find the equation in standard form of a parabola
with vertex at the origin and focus (5, 0).

1 2
x y
20
Ellipse
•Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy
Adams, while a member of the House of Representatives, discovered this acoustical
phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily
eavesdropping on the private conversations of other House members located near the
other focal point.
Ellipse center (0, 0)
•
•
•
•
•
•
Major axis - longer axis contains foci
Minor axis - shorter axis
Semi-axis - ½ the length of axis
Center - midpoint of major axis
Vertices - endpoints of the major axis
Foci - two given points on the major axis
Focus
Center
Focus
What is in an Ellipse?
• The set of all points in the plane, the sum of whose
distances from two fixed points, called the foci, is a
constant. (“Foci” is the plural of “focus”, and is
pronounced FOH-sigh.)
Why are the foci of the ellipse
important?
• The ellipse has an important property that is
used in the reflection of light and sound waves.
Any light or signal that starts at one focus will
be reflected to the other focus. This principle is
used in lithotripsy, a medical procedure for
treating kidney stones. The patient is placed in
a elliptical tank of water, with the kidney stone
at one focus. High-energy shock waves
generated at the other focus are concentrated
on the stone, pulverizing it.
Why are the foci of the ellipse
important?
• St. Paul's Cathedral in London. If a person
whispers near one focus, he can be heard at the
other focus, although he cannot be heard at many
places in between.
© 1994-2004 Kevin Matthews and Artifice, Inc. All Rights Reserved.
Ellipse
(x  h) 2 (y  k) 2

1
2
2
a
b
(x  h) 2 (y  k) 2

1
2
2
b
a
Center (h, k)
Major Axis (longer axis)
Horizontal
Minor Axis (shorter axis)
Vertical
Major Axis (longer axis)
Vertical
Minor Axis (shorter axis)
Horizontal

Foci (located on major axis)
F1 (c, 0) F2 (-c, 0)
b c  a
2
2
Foci (located on major axis)
F1 (0, c) F2 (0, -c)
2
OR
b  a c
2
2
2
Ellipse
• The ellipse with a center at (0,0) and a horizontal axis
has the following characteristics……
• Vertices (4, 0) and (-4, 0)
• Co-Vertices (0, 3) and (0, -3)
• Foci  7, 0and  7, 0
x2 y2

1
16 9
Ellipse
• The ellipse with a center at (0,0) and a vertical axis has
the following characteristics……
• Vertices (0, 9) and (0, -9)
• Co-Vertices (3, 0) and (-3, 0)
• Foci 0, 6 2 and 0,  6 2 
x2 y2

1
9 81
Ellipse
• The ellipse with a center at (-2, 2) and a horizontal
axis has the following characteristics……
(x  2) 2 (y  2) 2

1
16
9
• Vertices (2, 2) and (-6, 2)
• Co-Vertices (-2, -1) and (-2, 5)
• Foci 2  7, 2and 2  7, 2
Ellipse
• The ellipse with a center at (3, -1) and a vertical axis
has the following characteristics……
(x  2) 2 (y  2) 2

1
36
64
• Vertices (3, 7) and (3, -9)
• Co-Vertices (9, -1) and (-3, 1)
• Foci 3, 1 2 7 and 3, 1  2 7 

25x 2  4y 2 100x  8y  4
Graph this equation:

Standard
Center:
Foci:

(-2, 1)
2,1 
Vertices:

(x  2) 2 (y 1) 2
Form:

1
4
25

21
(0, 1) (-4, 1)
(-2, -4) (-2, 6)
Graph this equation:
9x 2  4 y 2 18x 16y 11

Standard
Center:
Foci:
Vertices:

(x 1) 2 (y  2) 2
Form:

1
4
9

(1, -2)
1,  2  5 
(-1, -2) (3, -2)
(1, -5) (1, 1)
Problems
Graph
4x 2 + 9y2 = 4
Find the vertices and foci of an ellipse:
sketch the graph
4x2 + 9y2 – 8x + 36y + 4 = 0
put in standard form
find center, vertices, and foci
Write the equation of the ellipse
• Given the center is at (4, -2) the foci are
(4, 1) and (4, -5) and the length of the
minor axis is 10.
(x  4) (y  2)

1
25
34
2

2
Hyperbolas
The huge chimney of a nuclear power plant
has the shape of a hyperboloid, as does
the architecture of the James S. McDonnell
Planetarium of the St. Louis Science
Center.
What is a Hyperbola?
• The set of all points in the plane, the
difference of whose distances from two fixed
points, called the foci, remains constant.
Where are the Hyperbolas?
•
A sonic boom shock wave has the
shape of a cone, and it intersects
the ground in part of a hyperbola.
It hits every point on this curve at
the same time, so that people in
different places along the curve on
the ground hear it at the same
time. Because the airplane is
moving forward, the hyperbolic
curve moves forward and
eventually the boom can be heard
by everyone in its path.
© Jill Britton, September 25, 2003
Hyperbola
(x  h) 2 (y  k) 2

1
2
2
a
b
(y  k) 2 (x  h) 2

1
2
2
a
b
c 2 = a2 + b2
Center (h, k)

Transverse axis
length = 2a
Vertices (a, 0) and (-a, 0)
Foci (c, 0) and (-c, 0)
(inside hyperbola)
Transverse axis
length = 2a
Vertices (0, a) and (0, -a)
Foci (0, c) and (0, -c)
(inside hyperbola)
Conjugate Axis
length = 2b
Conjugate Axis
length = 2b
Asymptotes y  
b
x
a
Asymptotes y  
a
x
b
Hyperbola
The Hyperbola with a center at (0,0) and a horizontal axis
has the following characteristics……
x2 y2

1
16 9
Vertices (4, 0) and (-4,0)
Foci (5, 0) and (-5, 0)

Asymptotes:
3
y  x
4
Hyperbola
The Hyperbola with a center at (0,0) and a vertical axis
has the following characteristics……
y2 x2

1
25 4
Vertices (0, 5) and (0, -5)
Foci 0, 29and 0,  29

Asymptotes:
5
y x
2
Hyperbola
The Hyperbola with a center at (3, -1) and a horizontal
axis has the following characteristics……
(x  3) 2 (y 1) 2

1
36
25
Vertices (9, -1) and (-3, -1)
Foci 3 61, 1and 3  61, 1
5
Asymptotes: y   x

6

Hyperbola
The Hyperbola with a center at (-2, 5) and a vertical axis
has the following characteristics……
(y  5) 2 (x  2) 2

1
49
25
Vertices (-2, 12) and (-2, -2)
Foci 2, 5  74 and 2, 5  74 
7
Asymptotes: y   x

5

Find each for this equation:
49y 2  36x 2 1764

Standard Form:
y2 x2

1
36 49
Center: (0, 0)

Foci:
0, 

85
Vertices: (0, 6) (0, -6)

Equations of
Asymptotes:
6
y x
7
Find each for this equation:
81x 2  36y 2  2916

Standard Form:
x2 y2

1
36 81
Center: (0, 0)

Foci:
3

13, 0
Vertices: (6, 0) (-6, 0)

Equations of
Asymptotes:
3
y x
2
Graph a hyperbola
Graph
2
2
y
x

1
25 36
Graph
x  62   y  32
25
144
1
Put into standard form
• 9y2 – 25x2 = 225
• 4x2 –25y2 +16x +50y –109 = 0
Write the equation of hyperbola
• Vertices (0, 2) and (0, -2)
Foci (0, 3) and (0, -3)
• Vertices (-1, 5) and (-1, -1)
Foci (-1, 7) and (-1, 3)
More Problems
Identify the graphs
•
•
•
•
•
•
4x2 + 9y2-16x - 36y -16 = 0
2x2 +3y - 8x + 2 =0
5x - 4y2 - 24 -11=0
9x2 - 25y2 - 18x +50y = 0
2x2 + 2y2 = 10
(x+1)2 + (y- 4) 2 = (x + 3)2

Classify each conic then write the
standard form of each equation:
1.) x2 + y2 – 4x – 12y + 30 = 0
2.) y2 – 4x2 – 2y – 16x = -1
CIRCLE
HYPERBOLA
(x  2)2  (y  6)2 10
(x  2)2  (y  6)2 10


Classify each conic then write the
standard form of each equation:
4.) 6x2 – 12x – 36 + 6y2 + 36y = 0
3.) 12x2 + 24x + 12y2 + 72y = 124
CIRCLE
CIRCLE
(x  2)2  (y  6)2 10
(x 1)2  (y  3)2 16


Classify each conic then write the
standard form of each equation:
5.) 4y2 – 3x + 7x2 – 7y = 18
6.) y2 + 9y = 7x + 3x2 + 52
ELLIPSE
HYPERBOLA
(x  2)2  (y  6)2 10
(x  2)2  (y  6)2 10
