MAT 129 - Ch 11 & 12

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MAT 129 – PrecalculusChapter 11& 12 NotesAnalytic Geometry, Conic Sections, & PDFDavid J. Gisch

Conics

Conic Sections

All the conic sections can be thought of as different slices (cross-sections) of the double cone.

Conic Sections

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Conic Sections

The Parabola

The Parabola The Parabola

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The ParabolaExample: Find an equation of the parabola with vertex at �0, 0�and focus at �3, 0�. Graph the equation.

The ParabolaExample: Find the equation of a parabola with vertex at �0, 0�and focus at �4, 0�. Graph the equation.

The ParabolaExample: The equation of a parabola is �� � 24�. Graph the equation and find the vertex and directrix.

The Parabola

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The ParabolaExample: Find the equation of a parabola with a focus at �0, 6�and directrix at � � � 6. Graph the equation.

The ParabolaExample: The equation of a parabola is �� � �8�, Graph the equation and state the focus, vertex and directrix.

The ParabolaExample: Find the equation of a parabola with a vertex at �4, �3�and focus �4, 0�. Graph the equation.

The ParabolaExample: The equation of a parabola is �� � 4� � 4� � 0. Graph the equation and state the focus, vertex, and directrix.

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Special Properties of the Parabola Special Properties of a Parabola

The ParabolaExample: A satellite dish measures 8 feet across at its opening and is 3 feet deep at its center. Where should the receiver be placed?

Example 11.2.8 Cont.

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The Ellipse

The Ellipse

An ellipse is a collection of all points in the plane, the sum of whose distances from two fixed points, called the foci, is constant. OR. For any point P in the plane and foci �� and �� the equation

� ��, � � � ��, � � �,where c is a constant, must be satisfied.

The Ellipse The Ellipse• The foci, center, and vertices on the minor axis form a right triangle.

This allows you to use trig and the Pythagorean theorem to find missing values.

• Also notice that the hypotenuse is half the length of the major axis and b is half the minor axis.

Major horizontal axis, equation��

�� ���

�� � 1, ����� � �� � ��

Major horizontal axis, equation��

�� ���

�� � 1, ����� � �� � ��

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The ParabolaExample: Find an equation of an ellipse centered at the origin with a focus of �4, 0�and vertex at �8, 0�. Graph the equation.

The ParabolaExample: Find the equation of an ellipse centered at the origin with a major axis of length 10 and a minor axis of length 6. Graph the equation.

The ParabolaExample 11.3.3: The equation of an ellipse is

��

36 ���

9 � 1

Find the foci, center, and vertices. Graph the ellipse.

The Ellipse

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The ParabolaExample: Find an equation of an ellipse centered at 2, 3 , with a focus of �2, 5�and a major axis of 6. Graph the equation.

The ParabolaExample: Find an equation of an ellipse centered at �1, 2 , with a focus of �0, 2�and a vertex at �0, 5�. Graph

the equation.

The EllipseExample: The equation of an ellipse is

4�� � �� � 32�� 4� � 52 � 0

Find the foci, center, and vertices. Graph the ellipse.

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Special Properties of the Ellipse The ParabolaExample: The whispering gallery in the Museum of Science and Industry in Chicago is 47.3 feet long. The distance from the center of the room to the foci is 20.3 feet. Find an equation that describes the shape of the room. How high is the room at its center?

Whispering Room Cont.

Parametric Equations

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Parametric Equations

We essentially break an equation into two pieces; one that models the x values, and one that models the y values. In doing so, each part is a function but when combined we can graph curves that are not normally functions, like the conics.

The ParabolaExample: Graph the parametric equation

� � � 3�, � � � 2��, �2 � �� 2

� � �

�2

�1

0

1

2

The ParabolaExample: Graph the parametric equation

� � � 3 cos �, � � � 3 sin �, �� � �� �

� � �

��

��2

0

�2�

Uses of Parametric Equations• One fun use of parametric equations is the science of projectiles.

• Here we can model a projectile with the parametric equations

� � � �� cos � �, � � � �12

��� � �� sin � �� �

where � is the force of gravity (32 �� ��⁄ or 9.8 � ��⁄ ), � is the angle to the horizontal, � is the height from the “ground,” and �� is the initial velocity of the projectile.

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Parametric EquationsExample: Including her height, Mara is on a cliff that is 150 feet high. She decides to become a drunk and get heavily involved in drugs and thus throws her dreams off of the cliff at an angle of 38° and a velocity of 25 feet per second.

a) Write a set of parametric equations modeling the flight of the dreams.

b) Graph the equations on a TI calculator

c) Determine the length of time that her dreams will be in the air.

d) How far did her dreams fly horizontally?

e) At what time will her dreams be at their peak height and what is that height?

Mara’s Dreams

Mara’s Dreams

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Parametric EquationsExample: An M16 machine gun has a muzzle velocity of 3,110 �� �⁄ . You are standing on level ground and shoot a bullet with an angle of 30° with the horizontal.

a) Write a set of parametric equations modeling the flight of the bullet.

b) Graph the equations on a TI calculator

c) Determine the length of time that the bullet will be in the air.

d) How far did her bullet fly horizontally?

e) At what time will the bullet be at its peak height and what is that height?

The Bullet

The Bullet Angry Projectiles

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Rectangular to Parametric EquationsExample: Find the parametric equation of

� = �� − 4

Rectangular to Parametric EquationsExample: Find the parametric equation of

�� +��

16 = 1

Parametric Equations Rectangular to Parametric EquationsExample: In the quintessential movie moment, you and your adversary, who are 15 feet apart, both see the object of supreme importance to completing the climax of the plot. Suddenly the lights go out. Your adversary takes off at a heading of 87° at 4 �� �⁄ . At the same time, you take off at a heading of 97° at 4.2 �� �⁄ .

a) Write a parametric equation for each persons movement.

b) Assuming you both were aiming correctly for the object, what are its coordinates?

c) Who gets to the object first? And at what time?

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Movie Moment

Rectangular to Parametric EquationsExample: A boat leaves the port on a heading of 45° going 40 knots per hour. At the same time a another boat takes off from a port 40 knots east of the first port, at a heading of 100° at 60 knots per hour.

a) Write a parametric equation for each boats movement.

b) Will the boats hit each other?

c) If yes, when? If not, when will they be the closest (to the nearest half hour)?

Ship Happens

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Partial Fraction Decomposition

Fraction Decomposition• Consider adding two rational expressions

3� + 3 +

2� − 3 =

3(� + 3)

(� − 3)(� − 3) +

2(� − 3)

(� + 3)(� + 3)

=3 � − 3 + 2(� + 3)

(� + 3)(� − 3) =5�− 1

�� + � − 12

• Adding and subtracting rational expressions is fairly easy but going in the reverse direction is more challenging. We call this process fraction decomposition.

• Note, we can only do this if the rational function is PROPER.

Fraction Decomposition

3� � 5�� � 3��� � 7��� � 2��

��� � 3 �

��� � 7 �

��� � 2

Method 1• ���� ��

��� ����= �

�� �+ �

���

• ���� ����� ����

(���)(�� �)�

= ����

(���)(�� �)�

+ ����

(���)(���)�

• − 3� − 17 = � � − 1 + � � + 3

• − 3� − 17 = �� − � + �� + 3�

• − 3� − 17 = �� + �� − � + 3�

• − 3� − 17 = � + � � − � + 3�

• Therefore, − 3 = � + �and − 17 = − � + 3�. This is a system of equations.

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Method 2• ���� ��

��� ����= �

�� �+ �

���

• ���� ����� ����

(���)(�� �)�

= ����

(���)(�� �)�

+ ����

(���)(���)�

• − 3� − 17 = � � − 1 + � � + 3

• Substitute � = 1,▫ − 3(1) − 17 = � 1 − 1 + � 1 + 3▫ − 20 = 4�▫ � = − 5

• Substitute � = − 3,▫ − 3(− 3) − 17 = � − 3 − 1 + � − 3 + 3▫ − 8 = − 4�▫ � = 2

Fraction decompositionExample: Decompose the rational function.

�(���)�������

Fraction decompositionExample: Decompose the rational function.

��������

Fraction decompositionExample: Decompose the rational function.

���������(���)(�������)

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Fraction Decomposition

3� � 5�� � 2��� � 1�� �

��� � 2 �

��� � 1 �

���� � 1�� �

���� � 1��

Fraction decompositionExample: Decompose the rational function.

����(���)(���)�

Fraction decompositionExample: Decompose the rational function.

��������