9.1
Circles and Parabolas
Objectives:

Recognize a _____________________ as the intersection of a plane and a double-napped cone

Write _______________________ of circles in standard form.

Write equations of ___________________________ in standard from.

Use circles and parabolas to model and solve _______________________________ problems.
Personalized Objective:

Circles
A Circle is the set of all points (x,y) in a plane that are equidistant
from a fixed point (h,k), called the center of the circle. The distance r
between the center and any point (x,y) on the circle is the radius.
The standard form of the equation of a circle is:
x  h 2   y  k 2  r 2
The point (h,k) is the center of the circle, and the positive number r is
the radius of the circle. The standard form of the equation of a circle
whose center is the origin, (h,k) = (0,0), is:
x2  y2  r 2
Name 5 places that you've seen circles in the world:

Finding the Standard Equation of a Circle
Example 1-pg. 661
The point (1,4) is on a circle whose center is at (-2,-3), write the standard form of the equation of the circle.
Summarize Steps:
Show work here:

Sketching a Circle
Example 2-pg. 662
Sketch the circle given by the equation
x 2  6 x  y 2  2 y  6  0 and identify the center and radius.
Summarize Steps:

Show work here:
Finding the Intercepts of a Circle
Example 3-pg. 662
Find the x- and y- intercepts of the graph of the circle given by the equation  x  4   y  2  16 .
2
Summarize Steps:

2
Show work here:
Application
You install a sprinkler that covers a circle with a radius of 30 feet. Let the sprinkler be at (0, 0).
a. Find the equation of the outer boundary of the sprinkler.
b. Your dog is standing 9 feet to the left and 27 feet closer to the house than the sprinkler. Will he get wet when
the sprinkler turns on (if he doesn't move)? Draw a picture to illustrate the situation.

Parabolas
--In this section we will keep our study of parabolas to be centered at the origin ( 0, 0 )
A Parabola is the 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
same line. The midpoint between the focus and the directrix is the
vertex, and the line passing through the vertex is the axis of the
parabaola.
The standard form of the equation of a parabola with center at (0, 0) is:
Equation
x  4 py
2
y 2  4 px
Focus
 0, p
 p, 0
Directrix
y  p
Vertical Axis (x = 0)
x  p
Horizontal Axis (y = 0)
What are some applications of parabolas?
Graph the two equations with p = 2 and p = -2, label all parts. There will be four graphs.

Write the parabola in standard form with the given information
Example 4 pg. 664 and other practice problems
A.
Vertex (0,0) and Focus (0,4)
B.
Vertex at origin and focus (3/2,0)
C.
Vertex (0,0) and directrix y=3
D.
Vertex (0,0) and directrix x=-3
What equation do you use for parabolas that open up or down?___________________________
What equation do you use for parabolas that open left or right?___________________________
Match the equation with its graph.
1.
y 2  2x
2.
x2  2y
3.
x 2  2 y
Example not in book- like #’s 55-59
Find the vertex, focus and directrix of
Summarize Steps:

y  2x 2 .
Show work here:
Applications
A.
A giant satellite dish is in the shape of a parabolic surface. Signals strike the surface and are
reflected to the focus, where the receiver is located. The diameter of the dish is 300 feet and its
depth is 44 feet. How far, to the nearest foot, from the base of the dish should the receiver be
placed?
B.
The towers of a suspension bridge are 800 feet apart and rise 160 feet above the road. The
cable between the towers has the shape of a parabola, and the cable just touches the sides of the
road midway between the towers. What is the height of the cable 100 feet away from a tower?
9.2
The Ellipse
Objectives:

Write equations of _________________________ in standard form.

Use properties of ellipses to _________________ ________ ________________ real-life problems.
An Ellipse is the set of all points (x, y) in a plane, the sum of whose distances between two fixed points is
constant. The two fixed points are called foci (pl) focus (singular). The line through the foci, intersects the
ellipse in two points called the vertices. The line segment joining the two vertices is called the major axis
The midpoint of the major axis is the center of the ellipse. The line segment perpendicular to the major axis at
the center of the ellipse is the minor axis. The endpoints of the minor axis are the co-vertices. The
eccentricity of an ellipse measures the ovalness of an ellipse. The eccentricity is always between zero and
one with smaller values producing a more circular ellipse.
Standard Form of an Ellipse
(Center at the Origin)
The standard form of the equation of an ellipse with center at (0, 0) and major and minor axes of lengths
2a and 2b where a  b is
x2 y2

1
a 2 b2
x2 y2

1
b2 a 2
Horizontal Major Axis
Vertical Major Axis
The vertices lie on the major axis, a units from the center.
The co-vertices lie on the minor axis, b units from the center.
The foci of the ellipse lie on the major axis, c units from the center, where
The eccentricity is given by the ratio e 
c
a
Where are ellipses seen or used in applications?
c2  a 2  b2 .

Finding the Standard Equation of an Ellipse.
Similar to Example 1 pg. 673 but centered at the origin. Find the standard form of the equation of the ellipse
having foci (-2,0) and (2,0) and a major axis of length 6. Also, state the eccentricity of the ellipse.
Summarize Steps:
Show work here:
You Try: Find the standard form of the equation of the ellipse having foci (-2,0) and (2,0) and a major axis of
length 12. Also, state the eccentricity of the ellipse.
Solution:

Sketching an Ellipse
Example 2 pg. 673
Sketch the ellipse given by
Summarize Steps:
4 x 2  y 2  36 and identify the center, vertices and eccentricity.
Show work here:
How can an ellipse be graphed on a calculator?
You try:
A.
9 x 2  16 y 2  144
B.
25 x 2  4 y 2  100
Match the equation with its graph.
1.
x2 y2

1
36 4
2.
x2 y2

1
9 36
3.
x2 y2

1
4 16

Finding the Equation of an Ellipse from its Foci or Eccentricity and Vertices
An example is not in the text.
Find the standard form of the equation of an ellipse with foci at (-1,0) and (1,0) and vertices (-2,0) and (2,0).
Summarize Steps:
Show work here:
You Try:
Find the standard form of the equation of an ellipse with foci at (-2,0) and (2,0) and vertices (-3,0) and (3,0).
Solution:
An example is not in the text.
Find the equation of the ellipse with vertices
Summarize Steps:

0,  8 and eccentricity e  12 .
Show work here:
Applications
A. A semi-elliptical archway is to be formed over the entrance to
an estate. The arch is to be set on pillars that are 10 feet apart.
The arch has a height of 4 feet above the pillars. Where
should the foci be placed in order to sketch plans for the
elliptical arch ?
B. A semielliptic archway has a height of 20 feet and a width of 50 feet. Can a truck 14 feet high and 10 feet
wide drive under the archway without going into the other lane?
9.3
The Hyperbola
Objectives:

Write equations of ________________________ in standard form.

Find _____________________ of and graph hyperbolas.

Use ______________________ of hyperbolas to solve real-life problems.
A hyperbola is the set of all points (x, y) in a plane, such that the difference of the distances between two fixed
points (foci) is a positive constant. The hyperbola contains two disconnected parts called branches. The line
through the foci, intersects the hyperbola in two points called the vertices. The line segment joining the vertices
is called the transverse axis. The midpoint of the transverse axis is the center of the hyperbola. You will need
to notice the differences between the hyperbola and ellipse to easily distinguish the two.

Graph Hyperbolas Centered at the Origin.
Horizontal Transverse Axis
Standard Form of a Hyperbola
(Center at the Origin)
The standard form of the equation of a hyperbola with center
at (0, 0) is
x2 y2

1
a 2 b2
y2 x2

1
a2 b2
Horizontal Transverse Axis
Vertical Transverse Axis
The vertices and the foci are, respectively, a and c from the
center, and c2  a 2  b2 .
Vertical Transverse Axis
Asymptotes of a Hyperbola
(Centered at the Origin)
The asymptotes of a hyperbola with center at (0, 0) are
b
b
x and y   x
a
a
a
a
y  x and y   x
b
b
y
Where are hyperbolas seen or used in applications?
Horizontal Transverse Axis
Vertical Transverse Axis

Finding the Standard Equation of a Hyperbola
Similar to Example 1 pg. 681 but centered at the origin. Find the standard form of the equation of the
hyperbola with vertices
0,2 and foci 0,5 .
Summarize Steps:

Show work here:
Sketching a Hyperbola
Example 2 pg. 682 Sketch the hyperbola whose equation is
vertices, foci and asymptotes of the hyperbola.
Summarize Steps:
Show work here:
4 x 2  y 2  16 . Then find the center, vertices, co-
Graph each hyperbola. Then locate the center, vertices, foci and find the equation of the asymptotes.
A.
x2 y2

1
25 16
You Try
B.
y2 x2

1
16 9
Graph each hyperbola. Then locate the center, vertices, foci and find the equation of the asymptotes.
A.
25x 2  4 y 2  100
You Try
B.
6 y 2  3x 2  18
Match the equation with its graph.
1.
x2 y2

1
36 4
2.
y2 x2

1
4 36
3.
x2 y2

1
4 36

Write Equations of Hyperbolas in Standard Form.
Similar to Example 4 pg. 684 but centered at the origin. Write the equation of each hyperbola.
A. Vertices:  2, 0 , 2, 0
B. Vertices: 0,  4 , 0, 4

Asymptotes:
C.
E.
 
y  4 x
 4,0,4,0
Vertices:  2, 0, 2, 0
Foci:

Asymptotes:
D.
Vertices:

Co-vertices:
F.
 
y  3 x,
10 , 0
0, 

15


Application
Example 5 pg. 685
An explosion is recorded by two microphones that are 1 mile apart. Microphone A received the sound 2 seconds
before microphone B. Assuming sound travels at 1100 feet per second, where did the explosion occur?