# a3_conics_s.1-4.web by qingyunliuliu

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```									 Warm-up                3/17/09

1)Find the values of x for which
sinx = 0 is true.
2)Give the amplitude and
period of the function
y = 3sin4x.
3)Find sin(sin -1 0.6)
Did You Know?
Q. What do bulletproof vests, fire escapes, windshield
wipers, and laser printers all have in common?
A. All invented by women.
F: One roach can live on a piece of gum for 5 years.
Q. If you were to spell out numbers, how far would you
have to go until you would find the letter "A"?
A. One thousand
• When a coffee seed is planted, it takes five years to
yield it's first consumable fruit.
• The common goldfish is the only animal that can
see both infra-red and ultra-violet light.
• Tennessee is bordered by more states than any
other. The eight states are Kentucky, Missouri,
Arkansas, Mississippi, Alabama, Georgia, North
Carolina and Virginia.
• Des Moines has the highest per capita Jello
consumption in the U.S
• The Western-most point in the contiguous United
States is Cape Alava, Washington.
• There are only three animals with blue tongues, the
Black Bear, the Chow Chow dog and the blue-
tongued lizard.
Folders
Conics Pre-Test
Warm-up   3/18/09
Did you know?
• In 1886, Coca-cola was first served at a pharmacy in
Atlanta, Georgia for only five cents a glass. A pharmacist
named John Pemberton created the formula for Coca-cola
• Flamingos are able to fly at a speed of approximately 55
kilometers an hour. In one night they can travel about 600
km
• Men are able to read fine print better than women can
• On average, 150 couples get married in Las Vegas each day
• Spiders usually have eight eyes, but still they cannot see
that well
• In humans, the epidermal layer of skin, which consists of
many layers of skin regenerates every 27 days
• Camel's milk does not curdle.
• The ball on top of a flagpole is called the truck.
• People generally read 25% slower from a computer screen compared
to paper
• Certain female species of spiders such as the Australian crab spider,
sacrifice their bodies as a food source for their offspring
• One grape vine produce can produce about 20 to 30 glasses of wine.
• The Hubble telescope is so powerful that it is like pointing a beam of
light at a dime that is two hundred miles away.
• On average people fear spiders more than they do death
• Every day, over five billion gallons of water are flushed down toilets in
the United States
• In one trip, a honey bee visits about 75 flowers
• Jupiter is the fastest rotating planet, which can complete one
revolution in less than ten hours
• A chicken loses its feathers when it becomes stressed
Conics Pre-Test
Today’s lesson:

Geometry Review

By the end of the lesson today, you
should be able to:
• Find the distance and midpoint
between two points on a coordinate
plane.
• Prove geometric relationships among
points and lines using analytical
methods.
10.1: Analytic Geometry
Distance Formula vs. Deriving the
formula
Can use graphing techniques and
pythagorean theorem.
Mid-point formula:
Use the words to make sense of the
formula
Ex1
Find the distance between points (-2,8) and
(6,2).

10 units
Ex2
Two children are playing hide and seek on the
school playground. One of the children is hiding at
the location (-4,3) on a map grid of the playground.
The child who is doing the seeking is currently at
the location (5, -2). Each side of a square on the
playground grid represents 5 feet. How far apart
are the two children?

Map distance is 10.3 units; unit = 5 feet, so 51.5 feet.
Ex3
vertices A(5,3), B(4,-2),C(-1,-2), and D(0,3) is a
parallelogram.

*Parallelogram*
One pair of opposite sides are congruent and parallel.
The measures of the two sides are equal, and the
slopes are parallel, so ABCD is a parallelogram.
Ex4
Find the coordinates of the midpoint
of the segment that has endpoints
(4,8) and (5,3).

(1/2, 11/2)
Ex5
Prove that the diagonals of a rectangle are
congruent.

Use corners:
(0,0)
(a,0)
(0,b)
(a,b)
Lesson Overview 10-1A
Lesson Overview 10-1B
5-Minute Check Lesson 10-2B
Assignment

Summary Quiz
Hw:
p.620-621
#12-36 Even
Warm-up                            3/19/09
1) A circle has a radius of 12 inches. Find the degree
measure of the central angle subtended by an arc
11.5 inches long.
54.9 °
2) Find sin390°.
½ or 0.5
3) Solve z2 – 8z = -14 by completing the square.
4 ± √2
4) If x2 = 16 and y2 = 4, what is the greatest possible
value of (x - y)2?
36
Did you know?
• Sharks are immune to cancer
• Manicuring the nails has been done by people for
more than 4,000 years
• The study of the iris of the eye is called iridology
• Back in 1919, the Russian transplant pioneer Serge
testicles onto human males’.
• In 1946, the New York Yankees became the first
baseball team to travel by plane
• By recycling just one glass bottle, the amount of
energy that is being saved is enough to light a 100
watt bulb for four hours
• The Mall of America, located in Bloomington, Minnesota is
so big that it can hold 24,336 school buses
• If you have three quarters, four dimes, and four pennies,
you have\$1.19. You also have the largest amount of money
in coins without being able to make change for a dollar.
• In the United States, poisoning is the fourth leading cause
of death among children
• In 1916, Charlie Chaplin was making \$10,000 a week,
making him the highest paid actor of his time
• It's possible to lead a cow upstairs...but not downstairs.
• People that smoke have 10 times as many wrinkles as a
person that does not smoke
• Thomas Edison was afraid of the dark. (Hence, the light
bulb?)
Check, go over hw

10.1
p.620-621
#12-36 Even

Questions
By the end of today’s lesson, you
should be able to:
• Use and determine the
standard and general forms
of the equation of a circle
• Graph circles
From
Algebra
II…
Definitions
• Circle –
–Set of all pints in a plane an equal
distance from a center point.

–Distance from the center to any point
on the circle.

notice pictures, vocabulary, etc.
Equation of a circle:
If r represents the radius of a circle with its center at
the origin, then the equation of the circle can be
written in the form:
x2 + y2 = r2
The equation of the circle with center that is Not at
(0,0) but at another point (h,k) is:
(x – h)2 + (y – h)2 = r2
This is the standard form of the equation for a circle.
Application
A tree in your garden has a ring of flowers.
Each flower is four feet from the center of the
tree trunk. You draw a coordinate grid to
model your yard. The tree trunk is located at
(-4,3). Write an equation that models the ring
of flowers.
(x - -4) 2 + (y – 3)2 = 42
(x + 4)2 + (y – 3)2 = 16
Try This
1) Write the equation of a circle whose
center is at (5,-3) and whose radius
is 5.
2) Describe the translation that gives
you the equation (x + 2)2 + (y – 5)2 =
1.
Using the center and radius to
graph a circle
Find the center and radius of the
circle:
(x + 6) 2 + (y – 7) 2 = 25

Find the center and radius of the circle
whose equation is
(x – 16) 2 + (y + 9) 2 = 144
Describe how you could use a stake
and a long piece of rope to mark the
perimeter of a circular home that will
be 20 feet across.
Lesson Overview 10-2A
General Form of Circle Equation:
x2 + y2 + Dx + Ey + F = 0
When the equation of a circle is given
in general form, it can be rewritten in
standard form by completing the
square for the terms in x and the
terms in y.
Example…
Lesson Overview 10-2B
Example
The equation of a circle is
x2 + y2 + 4x – 6y + 4 = 0.

a.   Write the standard form of the
equation.

b.    Find the radius and the coordinates of
the center.
a. x2 + y2 + 4x – 6y + 4 = 0
Group to form perfect square trinomials.
(x2 + 4x + ?) + (y2 – 6y + ?)= -4
(x2 + 4x + 4) + (y2 – 6y + 9)     = -4 + 4 + 9Complete
the square.
(x + 2)2 + (y – 3)2 = 9Factor the trinomials.
(x + 2)2 + (y – 3)2 = 32

b.The center of the circle is located at (-2, 3) and the
Now, you try:
The equation of a circle is
2x2 + 2y2 – 4x + 12y – 18 = 0

Write the standard form of the
equation
Find the radius and the coordinates of
the center.
Finding the equation of a circle
that passes through three points.
• Substitute in the (x,y) points for x and y in the
general form of the equation
• Use matrices to determine what D,E,F are.
• Write the equation
• Use completing the square to simplify.
Ex)
Write the standard form of the equation of the
circle that passes through the points at (5,3),
(-2,2), and (-1, -5)
Plug in all three sets of points for x & y:
x2 + y2 + Dx + Ey + F = 0
(5)2 + (3) 2 + D(5) + E(3) + F = 0
(-2) 2 + (2) 2 + D(-2) + E(2) + F = 0
(-1) 2 + (-5) 2 + D(-1) + E(-5) + F = 0
SOLVE THE 3X3 MATRIX:
5D + 3E + F = -34
-2D + 2E + F = -8
-D – 5E + F = -26

D=-4, E = 2, F = -20
NOW, JUST PUT THOSE NUMBERS
INTO THE GENERAL FORM AND
SOLVE FOR THE CIRCLE:

x2 + y2 + Dx + Ey + F = 0
x2 + y2 + -4x + 2y + -20 = 0
AFTER COMPLETING THE SQUARE,
YOU GET:
(X – 2)2 + (Y + 1)2 = 25
Now, you try
Write the standard form of the
equation of the circle that passes
through the points (-2, 3), (6, -5),
and (0, 7). Then identify the center
and the radius of the circle.
Substitute each ordered pair for (x, y) in
x2 + y2 + Dx + Ey + F = 0, to create a system of
equations.
(-2)2 + (3)2 + D(-2) + E(3) + F = 0(x, y) = (-2, 3)
(6)2 + (-5)2 + D(6) + E(-5) + F = 0(x, y) = (6, -5)
(0)2 + (7)2 + D(0) + E(7) + F = 0 (x, y) = (0, 7)
Simplify the system of equations.
-2D + 3E + F + 13 = 0
6D – 5E + F + 61 = 0
7E + F + 49 = 0
The solution to the system is D = -10, E = -4, and F = -
21.
The equation of the circle is
x2 + y2 – 10x – 4y – 21 = 0.
After completing the square, the standard
form is (x – 5)2 + (y – 2)2 = 50.
The center of the circle is (5, 2)
and the radius is √50 or 5√2.
Assignment

Section 10.2
p. 628-630
#25,28,35-39all, 41a,
43, 48, 55, 58
15 problems total
Warm-up   3/20/09
Did you know?
• Just by recycling one aluminum can, enough energy would be
saved to have a TV run for three hours.
• The first telephone call from the White House was from
Rutherford Hayes to Alexander Graham Bell
• Turtles can breathe through their butts
• A glockenspiel is a musical instrument that is like a xylophone. It
has a series of metal bars and is played with two hammers
• Teenage suicide is the second cause of death in the state of
Wisconsin
• Diamonds were first discovered in the riverbeds of the Golconda
region of India over 4,000 years ago.
• French artist, Michel Vienkot, uses cow dung as paint when he
creates his pictures
• There are 122 pebbles per square inch on a Spalding basketball
• The seventeenth president of the United States, Andrew
Johnson did not know how to read until he was 17 years old
• The fastest growing tissue in the human body is hair
• A cubic yard of air weighs about 2 pounds at sea level.
• Pancakes are served for breakfast, lunch and dinner in Australia
• A lion feeds once every three to four days
• A honey bee has four wings
• Cheddar cheese is the best selling cheese in the USA
• In the movie "The Matrix Reloaded" a 17 minute battle scene
cost over \$40 million to produce
Assignment

Section 10.2
p. 628-630
#25,28,35-39all, 41a,
43, 48, 55, 58
15 problems total
§10.3: Ellipses
LEQ: How do you identify the
characteristics of an ellipse?

ellipse?
Review Words
Major Axis
Vertices

Focus                  Center

co-vertices   minor axis
Standard Form of an Ellipse
x 2 + y2 = 1          x2 + y2 = 1
a2 b 2                b 2 a2
If a > b              If b > a
Major axis:           Major axis:
horizontal(x)         vertical(y)
Vertices: (±a,0)      Vertices: (0,±a)
Co-Vertices:(0, ±b)   Co-Vertices:(±b,0)
The Foci
The foci are important points in an
ellipse. The foci of an ellipse are
always on the major axis and are c
units from the center.

C2 = a2 – b2
• EX2 Find the foci of the ellipse with the equation
25x2 + 9y2 = 225
• Divide the coefficients by 255 (must = 1)
X 2 + y2 = 1
9 25
• The major axis is y. Vertices are (0,5)(0,-5)
• The minor axis is x. Co-vertices are (3,0) (-3,0)
• Foci are on the major axis: using the formula
• C2 = a2 – b2 a2 is always the larger #
• C=4        so the foci are at (0,4) and (0,-4)
Eccentricity
• The eccentricity of an ellipse, “e”, is a measure
that describes the shape of an ellipse.
• “e” is defined as c÷a
• Since c is smaller than a, c/a is always a
fraction between 0 and 1.
• The closer “e” is to 0, the more it looks like a
circle
• The closer “e” is to 1, the more if looks like an
oval.
Ex1

Find the equation of the ellipse
with the given information.
Ex2

Find the equation of the ellipse
with the given information.
Ex3
SPACE The graph models the elliptical path
of a space probe around two moons of a planet.
The foci of the path are the centers of the
moons. Find the coordinates of the foci.
The center of the ellipse is the origin.
a = (314) or 157
b = (110) or 55

The distance from the origin to the foci is c km.
c2 + b2 = a2
c2 + 552 = 1572
c  147

The foci are at (147, 0) and (-147, 0).
Ex4
Consider the ellipse graphed below.

a.Write the equation of the ellipse in standard
form.

b.Find the coordinates of the foci.
a. The center of the graph is at (-2, 1). Therefore, h = -
2 and k = 1.
Since the ellipse’s vertical axis is longer than its
horizontal axis, a is the distance between points at
(-2, 1) and (-2, -3) or 4. The value of b is the distance
between points at (-2, 1) and (1, 1) or 3.

b.Using the equation c = , we find that c = . The foci
are located on the vertical axis, units from the
center of the ellipse. Therefore, the foci have
coordinates (-2, 1 - √7) and (-2, 1 + √7).
Ex5
Find the coordinates of the center, the foci, and
the vertices of the ellipse with the equation 9x2 +
16y2 + 54x – 32y – 47 = 0. Then graph the equation.

First, write the equation in standard form.
9x2 + 16y2 + 54x – 32y – 47 = 0
9(x2 + 6x + ?) + 16(y2 – 2y + ?) = 47 + ? + ?
9(x2 + 6x + 9) + 16(y2 – 2y + 1) = 47 + 9(9) + 16(1)
Complete the square.
9(x + 3)2 + 16(y – 1)2 = 144
Write in standard form and graph.
10.3 Summary Quiz

Assignment (10 problems)
p. 638-640
#18,24,26,34,36
38,49,52,53a,63
5-Minute Check Lesson 10-4A
5-Minute Check Lesson 10-4B
Warm-up                                       3/23/09
Find the coordinates of the vertex, foci, vertices, and sketch the graph.
1) 25x2 + 9y2 + 100x – 18y = 116
2) 9x2 + 4y2 – 18 + 16y = 11
Write the equation of the ellipse that satisfies the following information:
3) The center is at (-2,-3), the length of the vertical
major axis is 8 units , and the length of the minor
axis is 2 units.
4) The foci are located at (-1,0) and (1,0) and a = 4.
5) The center is at (1,2), the major axis is parallel to
the x-axis, and the ellipse passes through points at
(1,4) and (5,2).
Did you know?
• According to research, Los Angeles highways are so
congested that the average commuter sits in traffic for
82 hours a year
• Over one million Pet Rocks were sold in 1975, making
Gary Dahl, of Los Gatos, California, a millionaire. He got
the idea while joking with friends about his pet that
was easy to take care of, which was a rock
• Because metal was scarce; the Oscars given out during
World War II were made of plaster
• The game Monopoly has been played by approximately
500 million people in the world, and the game is
available in 26 languages
• In 1983, a Japanese artist, Tadahiko Ogawa, made a copy of
the Mona Lisa completely out of ordinary toast
• Average number of days a West German goes without
washing his underwear: 7
• Cotton crops can be sprayed up to 40 times a year making it
the most chemical-intensive crop in the world
• Every year in the U.S., there are 178,000 new cases of lung
cancer
• On average, the American household consumes six pounds
of peanut butter annually
• A housefly can only ingest liquid material. They regurgitate
their food to liquefy the food that they are going to eat
• You can send a postcard from Hell. There is a small town
located in the Cayman Islands called "Hell." They even have
a post office
10.3 Summary Quiz

Assignment (10 problems)
p. 638-640
#18,24,26,34,36
38,49,52,53a,63
10.4: By the end of the lesson
today, you should be able to:
• Use and determine the
standard and general forms of
the equation of a hyperbola
• Graph a hyperbola.
Lesson Overview 10-4A
Lesson Overview 10-4B
Drawing a Hyperbola

• Use the standard form

• Find the values of a and b

• Draw a central rectangle

• Draw the asymptotes (diagonals of square)

• Draw branches through vertices
Ex1) Graph 9x2 – 25y2 = 225
Re-write in standard form:
9x2 – 25y2 = 1
225 225
x 2 – y2 = 1
25 9              a = 5, b = 3
Transverse axis (major one) is under x
Vertices are (5,0) and (-5,0)
Use a and b to draw the “rectangle”
Draw the asymptotes and diagonals
Equation of diagonals (y’s/x’s): y = ±3/5x
Finding the Foci
The foci of a hyperbola follow a similar rule to
those of an ellipse.
However, it is exactly the same as the
“Pythagorean theorem”

c2 = a2 + b2
Ex2) Find the foci of the hyperbola x2 – y2 = 1
36 4
Transverse axis is x-axis.
a = 6, b = 2       Vertices are at (6,0) & (-6,0)
C = √(62 + 22)
C = √40 ≈ 6.32 foci are at (6.32, 0) & (-6.32,0)
Draw square, Draw diagonals,
Equation of asymptotes (y’s/x’s) y = ±2/6x
Draw curves through vertices and curving at
asymptotes.
Ex3)
Find the coordinates of the center, foci,
and vertices, and the equations of the
asymptotes of the graph, then graph the
equation.

(y + 4)2 – (x – 2)2 = 1
36        25 .
10.4 Summary Quiz

Assignment/HW:
p. 650-651
#21-25 odd, 31-41 odd, 45

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