Unit 4
Conics
Conic Sections are formed by the intersection
of a double right cone and a plane.
Circles
( x  h)2  ( y  k )2  r 2
Vocabulary: Parts of a Circle
1. Center: the point equidistant
from all points on the edge
2. Radius: The segment from the
center to the edge of a circle
3. Diameter: a chord that passes
through the center of a circle
4. Chord: a segment whose
endpoints are on the circle
5. Tangent: Intersects the circle at
exactly one point
6. Secant: a line that intersects the
circle at 2 points.
Example: center: (4, 3)
Radius: 2
Equation:
Domain:
Range:
Example: center: Origin
Radius: 5
Equation:
Domain:
Range:
y
-1
-2
-3
-4
-5
-6
-7
-8
-9
All points a set distance from a center
point.
Standard Form:
( x  h)2  ( y  k )2  r 2
Center: (h, k)
Radius: r
Graph:
1. Write the equation in standard form
2. Identify the center and the radius
3. Plot the center and use the radius to
find all points “r” units from the center
y
9
8
7
6
5
4
3
2
1
y
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
Circles
9
8
7
6
5
4
3
2
1
x
1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
x
1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
x
1 2 3 4 5 6 7 8 9
( x  h)
(y  k )

1
2
2
a
b
2
Ellipse
2
Ellipse
All points on a plane where the sum of the distance from each point to the foci is constant
Vertical Major Axis
Horizontal Major Axis
( x  h)
(y  k )

1
2
a
b2
2
2
(y  k )
( x  h)

1
2
a
b2
2
Center: (h, k)
Major axis is 2a. Each vertex is “a” units from center
Minor axis is 2b. Each co-vertex is “b” units from center
2
2
2
Foci are located “c” units from center where c  a  b
2
Center: (h, k)
Major axis is 2a. Each vertex is “a” units from center
Minor axis is 2b. Each co-vertex is “b” units from center
2
2
2
Foci are located “c” units from center where c  a  b
y
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
Graph:
x
1 2 3 4 5 6 7 8 9
1. Put in standard form
2. Find “a” (in an ellipse, “a” is always bigger)
3. Find “b” (use one of the following methods)
a. “b” is given
b. Focus “c” is given use c 2  a 2  b 2
c. A point is given, substitute and solve for
“b”
4. Determine orientation
Eccentricity
Ellipse is more circular
If
c
is closer to 0: Then, Foci are closer to center.
a
Ellipse is very elongated
If
c
is closer to 1: Then, Foci are closer to vertices.
a
( x  h)
(y  k )

1
a2
b2
2
Hyperbola
All points on a plane where the difference of the distance from
each point to the foci is constant.
2
Hyperbola
The Fundamental Rectangle: The diagonals of the fundamental rectangle
extended in either direction are the asymptotes of the hyperbola.
Vertical Transverse Axis
How to graph hyperbolas:
(y  k )
( x  h)

1
2
a
b2
2
2
Center: (h, k)
1.
2.
3.
4.
5.
Transverse Axis is 2a units. Vertices are “a”
units from center
Conjugate Axis is 2b units. Co-vertices are “b”
units from center
6.
7.
Write the equation in standard form
Find Center
Determine the Transverse Axis. (“a” always comes first)
Find vertices
Draw the Fundamental Rectangle
a. Plot the vertices “a” units from center
b. Plot points on conjugate axis “b” units from center
c. Draw lines through each vertex (hor/ver)
d. Draw lines through each point (hor/ver)
Draw Asymptotes (diagonals)
Sketch Graph
Foci are “c” units from center where
c 2  a2  b2
y
( x  h)
(y  k )

1
2
a
b2
2
2
9
8
7
6
5
4
3
2
1
Horizontal Transverse Axis
Center: (h, k)
Transverse Axis is 2a units. Vertices are “a”
units from center
Conjugate Axis is 2b units. Co-vertices are “b”
units from center
Foci are “c” units from center where
c 2  a2  b2
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
x
1 2 3 4 5 6 7 8 9
2
(
x

h
)
 4 p( y  k ) Parabola
Parabola
All points the same distance from a fixed line (directrix) and a fixed point (focus).
Equation with vertex (h, k,)
Equation with vertex (h, k,)
( y  k )  4 p( x  h)
Direction parabola opens
Right
( x  h)  4 p( y  k )
Direction parabola opens
Up
( y  k )  4(  p)( x  h)
Left
( x  h)2  4(  p)( y  k )
Down
2
2
2
Vertex: (h, k)
Vertex: (h, k)
Focus: “p” units from vertex inside parabola
Focus: “p” units from vertex inside parabola
Directrix: a vertical line “p” units from vertex outside parabola
Directrix: a horizontal line “p” units from vertex outside parabola
y
Graph
1. Plot the vertex of the parabola (h, k)
2. Find the direction the parabola opens by using the tables above
3. Sketch the axis of symmetry.
a. Use the vertex and focus
b. The focus is “p” units away from the vertex in the direction the parabola opens.
c. The directrix is a vertical or horizontal line “p” units away from the vertex opposite from the focus
4. Find two reference points 2p units above/below or left/right of the focus based on its orientation.
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
x
1 2 3 4 5 6 7 8 9
Examples
y
Circles
A circle is centered at (-3, 5) and has a circumference of 8
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
Standard Form
2 x 2  2y 2  4 x  16 y  10  0
6 x 2  6 y 2  5 x  2y  7  0
x
1 2 3 4 5 6 7 8 9
-1
-2
-3
-4
-5
-6
-7
-8
-9
x 2  y 2  14 x  20 y  6  0
3 x 2  3 y 2  27  0
Parabolas
y
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
An ellipse centered at (-2, 3) with major axis length of 8 and
parallel to the y-axis, minor axis length 2
9 x 2  6 x  4 y  2  0
4 x 2  6 x  2y  26  0
x
1 2 3 4 5 6 7 8 9
-1
-2
-3
-4
-5
-6
-7
-8
-9
y 2 3 x  6 y  9  0
 y 2  x  2y  27  0
Ellipses
5 x 2  2y 2  10 x  8y  10  0
y
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
A vertical hyperbola with conjugate axis length 10, center
(2, -1) and one vertex at (2, 5)
8 y 2  x 2  14 x  6 y  6  0
15y 2  10 x 2  45 x  25y  8  0
x
1 2 3 4 5 6 7 8 9
-1
-2
-3
-4
-5
-6
-7
-8
-9
7 x 2  9 y 2  x  2y  4  0
Hyperbolas
10 x 2  14y 2  6 x  2y  100  0
 x 2  4y 2  16y  32  0
y
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
A parabola with focus at (4, 0) and directrix x=-4
x
1 2 3 4 5 6 7 8 9
9 x 2  2y 2  27 x  16 y  54  0
1.5 x 2  4y 2  4.5 x  8 y  20  0
Complete the Square
1. Complete the square of x 2  y 2 - 6x  2y - 6  0
2. Complete the square of 4 x 2  25y 2 - 40 x  100y  100  0
3. Complete the square 3y 2  4 x 2  12y  24 x  36
4. Complete the square y 2  2y  2x  7  0