Chapter 9 Topics in Analytical Geometry

1
Chapter 9 Topics in Analytical Geometry

• 9-1 Introduction to Conics Parabolas

2
Objective

• To prove theorems from geometry by using
  coordinates.

3
Introduction

• Analytical Geometry the study of geometric
  problems by means of analytical (or algebraic)
  methods
• Examples include the distance and midpoint
  formulas and equation of lines and parabolas.

4
Introduction

• Conic Sections are formed when two cones are
  placed tip to tip and extend infinitely in both
  directions.
• A circle, an ellipse, a hyperbola, and a parabola
  are all conic sections.

5
Warm-Up Exercises

• Complete the square for each of the following
• Solve the system of equations.

6
Introduction

• The circle is the first Conic Sections we will
  study. A circle is formed when the cone is
  sliced in a perpendicular fashion to the axis
  running through the conic section.

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Circle Notation

• The set of all points in the plane
  that are 5 units from the point is a
  circle.
• Use the distance formula to find the equation of
  this circle.

8
Circle Notation

• In general, if is on the circle with
• center and radius r.
• The center of this circle is and the
  radius is r. The equation of a circle with
  center at the origin and radius r is
  .

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Circles

• Examples
• Find the center and radius of each circle

10
Circles

• Examples
• Use a graphing calculator to graph the equation
  in the previous example.

11
Introduction

• The circle is the first Conic Sections we will
  study. A circle is formed when the cone is
  sliced in a perpendicular fashion to the axis
  running through the conic section.

12
Circle Notation

• The set of all points in the plane
  that are 5 units from the point is a
  circle.
• Use the distance formula to find the equation of
  this circle.

13
Circle Notation

• In general, if is on the circle with
• center and radius r.
• The center of this circle is and the
  radius is r. The equation of a circle with
  center at the origin and radius r is
  .

14
Circles

• Examples
• Find the center and radius of each circle

15
Circles

• Examples
• Use a graphing calculator to graph the equation
  in the previous example.

16
Warm-Up Exercises

• Consider the equation .
• Find the x and y-intercepts of the graph of the
  equation.
• Find the symmetry of the graph of the equation.
• Solve the equation for y.
• Write the equation that would translate the
  equation 2 units to the left and 3 units up.

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Introduction

• The parabola is the next Conic Sections we will
  study. We have previously studied parabolas.
  Parabolas opened up or down. This section will
  show that the

orientation of a parabola can be both vertical
and horizontal. A parabola is formed when the
cone is sliced with a plane that is parallel to
the outer edge of the conic section.
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Parabola Notation

• A parabola is the set of all points in a
  plane that are equidistant from a fixed line
  (directrix) and a fixed point (focus) not on the
  line.

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Parabola Notation

• The midpoint between the focus and the directrix
  is called the vertex, and the line passing
  through the focus and the vertex is called the
  axis of the parabola.

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Standard Equation

• The standard form of the equation of an parabola,
  with vertex is as follows.
• Vertical axis directrix
• Horizontal axis directrix

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Standard Equation

• The focus lies on the axis p units from the
  vertex. If the vertex is at the origin ,
  the equation takes one of the following forms.
• Vertical axis
• Horizontal axis

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Parabolas

• Example
• Find the standard form of the equation of the
  parabola with vertex and focus
  .
• Find the focus of the parabola

23
Parabolas

• Example
• Find the standard equation of the parabola with
  vertex at the origin and focus .

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Assignment

• Homework Assignment
• P. 218 Written Exercises 1-9 odd
• Homework Assignment (Day 2)
• P. 218 Written Exercises 11, 15-17