Conic Sections

Tim Brzezinski, Sep 13, 2015

Conic Sections: Circle, Ellipse, Parabola, & Hyperbola Resources

Table of Contents

  1. General
    1. Exploring Conic Section Possibilities
    2. Conic Sections: Introduction
    3. Conic Sections
  2. Circles
    1. Circle Equation: Center (0,0)
    2. Circle Equation: Center NOT (0,0)
    3. Circle Equation: Center (h,k)
    4. Equation & Graph of a Circle
    5. Equations of Circles (V1): Open Middle Problem
    6. Equations of Circles (V2): Open Middle Problem
    7. Equations of Circles (V3): Open Middle Problem
    8. Equations of Circles (V4): Open Middle Problem
  3. Parabola
    1. Parabola: Cause and Effect
    2. Locus Problem (1)
    3. Locus Construction 1
    4. Parabola as a Locus
    5. Parabola (Graph & Equation Anatomy)
    6. Special Conic LR Action
    7. Parabola: Geometric Property (I)
    8. Parabola: Geometric Property (II)
    9. Another Parabola Theorem!
    10. Parabolas: Quick Anatomy Exploration
    11. Reflectors, Cookers, and Other Phenomena
    12. Parabola (Locus): Quick Prompt
    13. Graphing Parabolas: Focus and Vertex Placement
    14. Quick Locus Exploration
  4. Ellipse
    1. Locus Problem (2)
    2. Locus Construction 2
    3. Quiz: Graphing Ellipses (EQ in Standard Form)
    4. Ellipse: Reflective Property
    5. Ellipse: Reflective Property
    6. Ellipse (Locus Construction)
    7. Ellipse (Graph & Equation Anatomy)
    8. Ellipse: Magic Sum = ?
    9. Ellipse: Magic Sum = ? (V2)
  5. Hyperbola
    1. Locus Problem (3)
    2. Locus Construction 3
    3. Hyperbola (Locus Construction)
    4. Hyperbola (Graph & Equation Anatomy)
    5. Hyperbola: Difference = ?
    6. Hyperbola: Reflective Property
    7. Another Conic?
    8. 2 Conic Sections
  6. Others
    1. Locus Constructions (via Paper Folding)

1.

General

  • 1. Exploring Conic Section Possibilities
  • 2. Conic Sections: Introduction
  • 3. Conic Sections

1.1.

Exploring Conic Section Possibilities

STUDENTS:
In each of the apps below, a [color=#bf9000][b]plane [/b][/color]intersects a [b][color=#1e84cc]double napped cone[/color][/b].
In the app above:
Note the equation of the plane is z = some constant. [br][br]Change the equation of this plane to [math]z=2[/math]. [br]Then change it to [math]z=1[/math]. [br]Then change it to [math]z=4[/math]. [br][br]How would you describe the intersection of this [b][color=#bf9000]plane[/color] [/b]and [color=#1e84cc][b]double-napped cone[/b][/color]?
In the app below, use the tools of GeoGebra to prove your assertion is correct.
Change the equation in the app below to create an entirely different cross section.
How would you describe this cross section in your own words? What does it look like?
Can you create an another cross section that's different from the other 2 examples above? Try to do so.
How would you describe this cross section in your own words? What does it look like?
Can you create a cross section different from the other three? Try to do so!
How would you describe this cross section in your own words? What does it look like?
Is it possible to create a cross section that is not curvy? How could we do it here?
What do you think?
When a plane intersects a double-napped circular cone, what could possible cross sections look like? [br]How would you describe them?
– Tim Brzezinski

1.2.

–

1.3.

–

2.

Circles

  • 1. Circle Equation: Center (0,0)
  • 2. Circle Equation: Center NOT (0,0)
  • 3. Circle Equation: Center (h,k)
  • 4. Equation & Graph of a Circle
  • 5. Equations of Circles (V1): Open Middle Problem
  • 6. Equations of Circles (V2): Open Middle Problem
  • 7. Equations of Circles (V3): Open Middle Problem
  • 8. Equations of Circles (V4): Open Middle Problem

2.1.

Circle Equation: Center (0,0)

For the questions below, be sure to zoom out if you need to!
1.
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius 5. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
2.
What is the equation of a circle with center (0,0) and radius [i]r[/i] = 9?
3.
Suppose another circle has center (0,0). Suppose this circle also passes through the point (12, -5).[br]Write the equation of this circle. [br]
4. FINAL QUESTION:
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius [i]r[/i], where [i]r[/i] > 0. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
Quick (Silent) Demo
– Tim Brzezinski

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

2.7.

–

2.8.

–

3.

Parabola

  • 1. Parabola: Cause and Effect
  • 2. Locus Problem (1)
  • 3. Locus Construction 1
  • 4. Parabola as a Locus
  • 5. Parabola (Graph & Equation Anatomy)
  • 6. Special Conic LR Action
  • 7. Parabola: Geometric Property (I)
  • 8. Parabola: Geometric Property (II)
  • 9. Another Parabola Theorem!
  • 10. Parabolas: Quick Anatomy Exploration
  • 11. Reflectors, Cookers, and Other Phenomena
  • 12. Parabola (Locus): Quick Prompt
  • 13. Graphing Parabolas: Focus and Vertex Placement
  • 14. Quick Locus Exploration

3.1.

Parabola: Cause and Effect

We know that the intersection of a plane and double-napped cone is a conic section. [br][br]One particular conic section shown below is a [b]PARABOLA[/b]. [br][br]Interact with the applet below for a few minutes. Then answer the questions that follow. [br][br][b][color=#0000ff]Note:[/color][/b][br][b]LARGE POINTS [/b]are moveable [b]AT ANY TIME[/b]. [br][br][b][color=#1e84cc]To explore in Augmented Reality, see the directions below the applet. [/color][/b]
1.
How would you describe the inclination of the [b][color=#38761d]green plane[/color][/b] with respect to the [color=#bf9000][b]cones[/b][/color]? Explain.
2.
How would you describe the action that happens towards the end of the animation?
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device. [br][br]2) Go to the MENU (horizontal bars) in the upper left corner. Select OPEN. [br] In the Search GeoGebra Resources input box, type [b]ztM2Yffm[/b][br] (Note this is the resource ID = last 8 digits of the URL for this resource.)[br][br][b]Note: [/b][br][br]The [b]i[/b] slider controls the animation. You can also move the [b][color=#bf9000]LARGE YELLOW POINT[/color][/b]. Then press AR again to explore in Augmented Reality.
– Tim Brzezinski

3.2.

–

3.3.

–

3.4.

–

3.5.

–

3.6.

–

3.7.

–

3.8.

–

3.9.

–

3.10.

–

3.11.

–

3.12.

–

3.13.

–

3.14.

–

4.

Ellipse

  • 1. Locus Problem (2)
  • 2. Locus Construction 2
  • 3. Quiz: Graphing Ellipses (EQ in Standard Form)
  • 4. Ellipse: Reflective Property
  • 5. Ellipse: Reflective Property
  • 6. Ellipse (Locus Construction)
  • 7. Ellipse (Graph & Equation Anatomy)
  • 8. Ellipse: Magic Sum = ?
  • 9. Ellipse: Magic Sum = ? (V2)

4.1.

Locus Problem (2)

In the applet below, [br][br][color=#1e84cc][b][i]O[/i] is the center [/b][/color]of the circle shown. [br][b]Point [i]D[/i] is a point that lies ON this circle. [/b][br][color=#1e84cc][b]Point [i]A[/i] is a point that ALWAYS LIES INSIDE[/b] [/color]the circle. (You can move it anywhere you'd like). [br]The [color=#ff00ff][b]pink line[/b][/color] is the [color=#ff00ff][b]perpendicular bisector[/b][/color] of the segment with endpoints [i]A[/i] and [i]D[/i]. [br][br]Drag [b]point [i]D[/i][/b] around the circle a few times. What do you see? Describe in detail! [br] [br]Feel free to alter the locations of [i][color=#1e84cc][b]A[/b][/color][/i] and [i][color=#1e84cc][b]R[/b][/color]. [/i]Then clear the trace and drag [b]point [i]D[/i][/b] around again. [br][br]Why does this occur?
Please go to the [url=https://www.geogebra.org/m/TZu6tRwE]Locus Construction 2 Task[/url] & begin!
– Tim Brzezinski

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

4.7.

–

4.8.

–

4.9.

–

5.

Hyperbola

  • 1. Locus Problem (3)
  • 2. Locus Construction 3
  • 3. Hyperbola (Locus Construction)
  • 4. Hyperbola (Graph & Equation Anatomy)
  • 5. Hyperbola: Difference = ?
  • 6. Hyperbola: Reflective Property
  • 7. Another Conic?
  • 8. 2 Conic Sections

5.1.

Locus Problem (3)

In the applet below, [br][br][color=#1e84cc][b][i]O[/i] is the center [/b][/color]of the circle shown. [br][b]Point [i]D[/i] is a point that lies ON this circle. [br][/b][color=#1e84cc][b]Point [i]A[/i] is a point that ALWAYS LIES OUTSIDE[/b] [/color]the circle. (You can move it anywhere you'd like). [br]The [color=#ff00ff][b]pink line[/b][/color] is the [color=#ff00ff][b]perpendicular bisector[/b][/color] of the segment with endpoints [i]A[/i] and [i]D[/i]. [br][br]Drag [b]point [i]D[/i][/b] around the circle a few times. What do you see? Describe in detail! [br] [br]Feel free to alter the locations of [i][color=#1e84cc][b]A[/b][/color][/i] and [b]the gray point (radius changer)[/b][i]. [br][/i]Then clear the trace and drag [b]point [i]D[/i][/b] around again. [br][br]Why does this occur?
Please go to the [url=https://www.geogebra.org/m/yXDC8N93]Locus Construction (3) Task[/url] and begin!
– Tim Brzezinski

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

6.

Others

  • 1. Locus Constructions (via Paper Folding)

6.1.

Locus Constructions (via Paper Folding)

Locus Construction (I)
[url=https://www.geogebra.org/m/sduBXC6P]Locus Construction (I)[/url] - [color=#0000ff]with GeoGebra[/color]
Locus Construction (II)
[url=https://www.geogebra.org/m/TZu6tRwE]Locus Construction (II)[/url] - [color=#0000ff]with GeoGebra[/color]
Locus Construction (III)
[url=https://www.geogebra.org/m/yXDC8N93]Locus Construction (III)[/url] - [color=#0000ff]with GeoGebra[/color]
– Tim Brzezinski

Information