Difference between revisions of "Applications-of-GeoGebra/C2/Conic-Sections-Ellipse/English"

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(Created page with "{|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Conic Sections - Ellipse''' |- | | '''Sl...")
 
 
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{|border=1
 
{|border=1
−
| | '''Visual Cue'''
+
|| '''Visual Cue'''
−
| | '''Narration'''
+
|| '''Narration'''
  
 
|-
 
|-
−
| | '''Slide Number 1'''
+
|| '''Slide Number 1'''
  
 
'''Title Slide'''
 
'''Title Slide'''
−
| | Welcome to this tutorial on '''Conic Sections - Ellipse'''
+
|| Welcome to this tutorial on '''Conic Sections - Ellipse'''
 
|-
 
|-
−
| | '''Slide Number 2'''
+
|| '''Slide Number 2'''
  
 
'''Learning Objectives'''
 
'''Learning Objectives'''
−
| | In this tutorial, we will learn,
+
|| In this tutorial, we will learn,
  
 
'''Standard equations''' and parts of an ellipse
 
'''Standard equations''' and parts of an ellipse
Line 18: Line 18:
 
To use '''GeoGebra''' to construct an ellipse
 
To use '''GeoGebra''' to construct an ellipse
 
|-
 
|-
−
| | '''Slide Number 3'''
+
|| '''Slide Number 3'''
  
 
'''System Requirement'''
 
'''System Requirement'''
−
| | Here I am using:
+
|| Here, I am using:
  
−
'''Ubuntu Linux '''OS version 14.04
+
'''Ubuntu Linux ''' OS version 14.04
  
 
'''GeoGebra 5.0.388.0-d'''
 
'''GeoGebra 5.0.388.0-d'''
 
|-
 
|-
−
| | '''Slide Number 4'''
+
|| '''Slide Number 4'''
  
 
'''Pre-requisites'''
 
'''Pre-requisites'''
−
 
−
 
−
 
−
 
−
 
  
 
'''www.spoken-tutorial.org'''
 
'''www.spoken-tutorial.org'''
−
| | To follow this '''tutorial''', you should be familiar with
+
|| To follow this '''tutorial''', you should be familiar with
  
 
'''GeoGebra''' interface
 
'''GeoGebra''' interface
Line 43: Line 38:
 
Conic sections in geometry
 
Conic sections in geometry
  
−
 
+
For relevant tutorials, please visit our website.
−
For relevant '''tutorials''', please visit our website.
+
 
|-
 
|-
−
| | '''Slide Number 5'''
+
|| '''Slide Number 5'''
  
 
'''Ellipse'''
 
'''Ellipse'''
  
−
[[Image:]]
+
An ellipse, is the locus of points whose sum of distances from two fixed points is constant.
−
 
+
−
An ellipse is the locus of points whose sum of distances from two fixed points is constant.
+
  
 
These fixed points are called the foci.
 
These fixed points are called the foci.
Line 63: Line 55:
  
 
Point to 2 latus recti passing through foci and the two axes.
 
Point to 2 latus recti passing through foci and the two axes.
−
| | An ellipse is the locus of points whose sum of distances from two fixed points is constant.
+
||'''Ellipse'''
 +
 
 +
An ellipse is the locus of points whose sum of distances from two fixed points is constant.
  
 
These fixed points are called the foci.
 
These fixed points are called the foci.
  
−
Observe the center '''O''', foci '''F1''' and '''F2'''.
+
Observe the centre '''O''', foci '''F1''' and '''F2'''.
  
−
Vertices '''A''' and '''B''' are at the ends of the major axis '''AB.'''
+
Vertices '''A''' and '''B''' are at the ends of the major axis '''AB'''.
  
 
'''Co-vertices C''' and '''D''' are at the ends of the minor axis '''CD'''.
 
'''Co-vertices C''' and '''D''' are at the ends of the minor axis '''CD'''.
Line 77: Line 71:
 
Axes lengths '''2a''' and '''2b''' and distance between the foci '''2c''' are shown in the figure.
 
Axes lengths '''2a''' and '''2b''' and distance between the foci '''2c''' are shown in the figure.
 
|-
 
|-
−
| |
+
||
−
| | Be careful to distinguish length from letters used for '''sliders''', circles and ellipses.
+
|| Be careful to distinguish length, from letters used for '''sliders''', circles and ellipses.
 
|-
 
|-
−
| | Show the '''GeoGebra''' window.
+
|| Show the '''GeoGebra''' window.
−
| | Let us construct an ellipse in '''GeoGebra'''.
+
|| Let us construct an ellipse in '''GeoGebra'''.
  
 
I have already opened the '''GeoGebra''' interface.
 
I have already opened the '''GeoGebra''' interface.
 
|-
 
|-
−
| | Click on '''Point''' tool and click twice in '''Graphics''' view.
+
|| Click on '''Point''' tool >> click twice in '''Graphics''' view.
−
| | Click on '''Point''' tool and click twice in '''Graphics''' view.
+
|| Click on '''Point''' tool and click twice in '''Graphics''' view.
 
|-
 
|-
−
| | Point to '''A''' and '''B'''.
+
|| Point to '''A''' and '''B'''.
−
 
+
|| This creates two points '''A''' and '''B''', which will be the foci of our ellipse.
−
| | This creates two points '''A''' and '''B''', which will be the foci of our ellipse.
+
 
|-
 
|-
−
| | Right-click on '''A''' and choose '''Rename''' option.
+
|| Right-click on '''A''' >> choose '''Rename''' option.
−
| | Right-click on '''A''' and choose the '''Rename''' option.
+
|| Right-click on '''A''' and choose the '''Rename''' option.
 
|-
 
|-
−
| | Type '''F1''' in the '''New Name''' field and click '''OK''' button.
+
|| Type '''F1''' in the '''New Name''' field >> click '''OK''' button.
−
| | In the '''New Name''' field, type '''F1''' and click '''OK'''.
+
|| In the '''New Name''' field, type '''F1''' and click '''OK'''.
 
|-
 
|-
−
| | Point to the focus '''F1'''.
+
|| Point to the focus '''F1'''.
−
| | This will be one of our foci, '''F1'''.
+
|| This will be one of our foci, '''F1'''.
 
|-
 
|-
−
| | Point to '''B'''.
+
|| Point to '''B'''.
−
| | Let us rename '''B''' as '''F2'''.
+
|| Let us rename '''B''' as '''F2'''.
 
|-
 
|-
−
| | Click on '''Slider''' tool >> click in '''Graphics''' view.
+
|| Click on '''Slider''' tool >> click in '''Graphics''' view.
−
| | Click on '''Slider''' tool and click in '''Graphics''' view.
+
|| Click on '''Slider''' tool and click in '''Graphics''' view.
 
|-
 
|-
−
| | Point to the '''Slider dialog box''' in '''Graphics''' view.
+
|| Point to the '''Slider dialog box''' in '''Graphics''' view.
−
| | '''Slider dialog box''' appears in '''Graphics''' view.
+
|| '''Slider dialog box''' appears in '''Graphics''' view.
 
|-
 
|-
−
| | Stay with the default '''Number''' selection and in the '''Name''' field, type '''k'''.
+
|| Stay with the default '''Number''' selection >> in the '''Name''' field, type '''k'''.
−
| | Stay with the default '''Number''' selection and in the '''Name''' field, type '''k'''.
+
|| Stay with the default '''Number''' selection and in the '''Name''' field, type '''k'''.
 
|-
 
|-
−
| | Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.
+
|| Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.
  
 
Click '''OK'''.
 
Click '''OK'''.
−
| | Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.
+
|| Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.
  
 
Click '''OK'''.
 
Click '''OK'''.
 
|-
 
|-
−
| | Point to '''slider k'''.
+
|| Point to '''slider k'''.
−
| | This creates a number '''slider''' named '''k'''.
+
|| This creates a number '''slider''' named '''k'''.
 
|-
 
|-
−
| | Drag '''slider k'''.
+
|| Drag '''slider k'''.
−
| | '''Slider k''' can be changed from 0 to 10.
+
|| '''Slider k''' can be changed from 0 to 10.
  
 
'''k''' will be the sum of the distances of any point on the ellipse from the foci '''F1''' and '''F2'''.
 
'''k''' will be the sum of the distances of any point on the ellipse from the foci '''F1''' and '''F2'''.
 
|-
 
|-
−
| |
+
||point to '''a'''.
−
| | We will create another '''number slider''' named “'''a'''”.
+
|| We will create another '''number slider''' named '''a'''.
 
|-
 
|-
−
| | Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.
+
|| Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.
  
 
Click '''OK'''.   
 
Click '''OK'''.   
−
| | Its '''Min''' value is 0, '''Max''' value is 10, '''increment''' is 0.1."
+
|| Its '''Min''' value is 0, '''Max''' value is 10, '''increment''' is 0.1."
 
|-
 
|-
−
| | Click on '''Circle with Center and Radius''' tool >> click on '''F1'''.
+
|| Click on '''Circle with Center and Radius''' tool >> click on '''F1'''.
−
| | Click on '''Circle with Center and Radius''' tool and click on '''F1'''.
+
|| Click on '''Circle with Center and Radius''' tool and click on '''F1'''.
 
|-
 
|-
−
| | Point to the '''text box'''; type '''a''' >> click '''OK'''.
+
|| Point to the '''text box'''; type '''a''' >> click '''OK'''.
−
| | A '''text-box''' appears; in the radius '''Name''' field, type '''a''' and click '''OK'''.
+
|| A '''text-box''' appears. In the '''Radius''' name field, type '''a''' and click '''OK'''.
 
|-
 
|-
−
| | Point to the circle with center '''F1''' and radius '''a'''.
+
|| Point to the circle with centre '''F1''' >> radius '''a'''.
−
| | A circle '''c''' with center '''F1''' and radius '''a''' appears.
+
|| A circle '''c''' with centre '''F1''' and radius '''a''' appears.
 
|-
 
|-
−
| | Drag '''slider a''' to 2 and '''slider k''' to 5.
+
|| Drag '''slider a''' to 2 >> '''slider k''' to 5.
−
| | Drag '''slider a''' to 2 and '''slider k''' to 5.
+
|| Drag '''slider a''' to 2 and '''slider k''' to 5.
 
|-
 
|-
−
| | Click again on '''Circle with Center and Radius''' tool >> click on '''F2'''.
+
|| Click on '''Circle with Center and Radius''' tool >> click on '''F2'''.
−
| | Click again on '''Circle with Center and Radius''' tool and click on '''F2'''.
+
|| Click again on '''Circle with Center and Radius''' tool and click on '''F2'''.
 
|-
 
|-
−
| | Point to the '''text box'''; type '''k-a''' and click '''OK'''.
+
|| Point to the '''text box'''; type '''k-a''' and click '''OK'''.
−
| | In the '''text box''' that appears, type '''k minus a''' and click '''OK'''.
+
|| In the '''text box''' that appears, type '''k minus a''' and click '''OK'''.
 
|-
 
|-
−
| | Point to circle '''d''' with center '''F2''' and radius '''k-a''' in '''Graphics''' view.
+
|| Point to circle '''d''' with center '''F2''' >> radius '''k-a''' in '''Graphics''' view.
−
| | A circle '''d''' with center '''F2''' and radius '''k minus a''' appears in '''Graphics''' view.
+
|| A circle '''d''' with center '''F2''' and radius '''k minus a''' appears in '''Graphics''' view.
 
|-
 
|-
−
| | Drag '''slider k''' to 5 and '''slider a''' to 2.
+
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
−
| | Drag '''slider k''' to 5 and '''slider a''' to 2.
+
|| Under '''Move Graphics View''', click on '''Zoom Out''' and in '''Graphics''' view.
 
|-
 
|-
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
+
|| Click on '''Move Graphics View''' >> drag '''Graphics''' view.
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
+
|| Click on '''Move Graphics View''' and drag '''Graphics''' view.
 
|-
 
|-
−
| | Click on '''Move Graphics View''' and drag '''Graphics''' view.
+
|| Click on '''Intersect''' tool under '''Point''' tool.
−
| | Click on '''Move Graphics View''' and drag '''Graphics''' view.
+
−
|-
+
−
| | Click on '''Intersect''' tool under '''Point''' tool.
+
  
 
Click on the '''intersections''' of the two '''circles'''.
 
Click on the '''intersections''' of the two '''circles'''.
−
| | Under '''Point''', click on '''Intersect''' tool.
+
|| Under '''Point''', click on '''Intersect''' tool.
  
 
Click on the two circles '''c''' and '''d'''.   
 
Click on the two circles '''c''' and '''d'''.   
 
|-
 
|-
−
| | Point to points '''A''' and '''B'''.
+
|| Point to points '''A''' and '''B'''.
−
| | This creates points''' A''' and '''B'''.
+
|| This creates points '''A''' and '''B'''.
 
|-
 
|-
−
| | Click on '''Segment''' tool under '''Line''' tool.
+
|| Click on '''Segment''' tool under '''Line''' tool.
−
| | Under '''Line''', click on '''Segment''' tool.
+
|| Under '''Line''', click on '''Segment''' tool.
 
|-
 
|-
−
| | Click on points '''F1''' and '''A''' to join them.
+
|| Click on points '''F1''' >> '''A''' to join.
−
| | Click on points '''F1''' and '''A''' to join them.
+
|| Click on points '''F1''' and '''A''' to join them.
 
|-
 
|-
−
| | Next, click on points '''A''' and '''F2''' to join them.   
+
|| Click on points '''A''' >> '''F2''' to join.   
−
| | Next, click on points '''A''' and '''F2''' to join them.
+
|| Next, click on points '''A''' and '''F2''' to join them.
 
|-
 
|-
−
| | Click on '''Move''' tool.
+
|| Click on '''Move''' tool.
−
| | Click on '''Move'''.
+
|| Click on '''Move'''.
 
|-
 
|-
−
| | Double click on '''Segment AF1'''.
+
|| Double click on '''Segment AF1'''.
−
| | Double click on '''Segment AF1'''.
+
|| Double click on '''Segment AF1'''.
 
|-
 
|-
−
| | Click '''Object Properties''' to open the '''Preferences''' dialog box.
+
|| Click '''Object Properties''' to open the '''Preferences''' dialog box.
−
| | Click on '''Object Properties''' to open the '''Preferences''' dialog box.
+
|| Click on '''Object Properties''' to open the '''Preferences''' dialog box.
 
|-
 
|-
−
| | Point to Segment '''AF1''' highlighted in the left panel.
+
|| Point to Segment '''AF1''' highlighted in the left panel.
−
| | Segment '''AF1''' is already highlighted in the left panel.
+
|| Segment '''AF1''' is already highlighted in the left panel.
 
|-
 
|-
−
| | Holding '''Ctrl''' key down, click and highlight Segment '''AF2'''.
+
|| Holding '''Ctrl''' key down, click >> highlight Segment '''AF2'''.
−
| | Holding '''Ctrl''' key down, click and highlight Segment '''AF2''' as well.
+
|| Holding '''Ctrl''' key down, click and highlight Segment '''AF2''' as well.
 
|-
 
|-
−
| | Under '''Basic''' tab >> select '''Show Label'''.
+
|| Under '''Basic''' tab >> select '''Show Label'''.
−
| | Under the '''Basic''' tab, make sure that '''Show Label''' is selected.
+
|| Under the '''Basic''' tab, make sure that '''Show Label''' is selected.
 
|-
 
|-
−
| | Pull down the '''drop down menu''' next to the '''Show Label''' check box.
+
|| Pull down the '''drop down menu''' next to the '''Show Label''' check box.
  
 
Select '''Name and Value'''.
 
Select '''Name and Value'''.
−
| | Pull down the '''drop down menu''' next to the '''Show Label''' check box.
+
|| Pull down the '''drop down menu''' next to the '''Show Label''' check box.
  
 
Select '''Name and Value'''.
 
Select '''Name and Value'''.
 
|-
 
|-
−
| | Under '''Color''' tab >> select Red.
+
|| Under '''Color''' tab >> select Red.
−
| | Under the '''Color''' tab, select red.
+
|| Under the '''Color''' tab, select red.
 
|-
 
|-
−
| | Under '''Style''' tab >> choose '''dashed line style'''.
+
|| Under '''Style''' tab >> choose '''dashed line style'''.
−
| | Under the '''Style''' tab, choose '''dashed line style'''.
+
|| Under the '''Style''' tab, choose '''dashed line style'''.
 
|-
 
|-
−
| | Close the '''Preferences''' dialog box.
+
|| Close the '''Preferences''' dialog box.
−
| | Close the '''Preferences''' dialog box.
+
|| Close the '''Preferences''' dialog box.
 
|-
 
|-
−
| | Follow earlier steps to draw Segments '''BF1''' and '''BF2'''.
+
|| Follow earlier steps to draw Segments '''BF1''' and '''BF2'''.
  
 
Make them '''dashed''' and blue.
 
Make them '''dashed''' and blue.
−
| | Draw Segments '''BF1''' and '''BF2'''.
+
|| Draw Segments '''BF1''' and '''BF2'''.
  
 
Make them '''dashed''' and blue.
 
Make them '''dashed''' and blue.
 
|-
 
|-
−
| | Point to '''Move''' tool.
+
|| Point to '''Move''' tool.
  
 
If not highlighted, click on it.
 
If not highlighted, click on it.
  
 
Move the labels so you can see them properly.
 
Move the labels so you can see them properly.
−
| | Make sure that the '''Move''' tool is highlighted.
+
|| Make sure that the '''Move''' tool is highlighted.
  
 
Move the labels so you can see them properly.
 
Move the labels so you can see them properly.
 
|-
 
|-
−
| | Point to the length labels next to '''AF1, AF2, BF1, BF2''' and '''slider k'''.
+
|| Point to the length labels next to '''AF1, AF2, BF1, BF2''' and '''slider k'''.
−
| | Note that the sum of the segment lengths from both foci to each intersection point is equal to '''k'''.
+
|| Note that the sum of the segment lengths from both foci to each intersection point is equal to '''k'''.
 
|-
 
|-
−
| | Right-click on '''A''' and '''B''' >> check '''Trace On''' option.
+
|| Right-click on '''A''' and '''B''' >> check '''Trace On''' option.
−
| | Right-click on '''A''' and '''B''' and check '''Trace On''' option.
+
|| Right-click on '''A''' and '''B''' and check '''Trace On''' option.
 
|-
 
|-
−
| | Uncheck circles '''c''' and '''d''' in '''Algebra''' view.
+
|| Uncheck circles '''c''' and '''d''' in '''Algebra''' view.
−
| | In '''Algebra''' view, uncheck circles '''c''' and '''d''' to hide the circles.
+
|| In '''Algebra''' view, uncheck circles '''c''' and '''d''' to hide the circles.
 
|-
 
|-
−
| | Right-click on '''slider a''' >> check '''Animation On''' option.
+
|| Right-click on '''slider a''' >> check '''Animation On''' option.
−
| | Right-click on '''slider a''' and check '''Animation On''' option.
+
|| Right-click on '''slider a''' and check '''Animation On''' option.
 
|-
 
|-
−
| | Right-click on '''slider k''' and check '''Animation On''' option.
+
|| Right-click on '''slider k''' >> check '''Animation On''' option.
−
| | Next, right-click on '''slider k''' and check '''Animation On''' option.
+
|| Next, right-click on '''slider k''' and check '''Animation On''' option.
 
|-
 
|-
−
| | Point to the traces of '''A''' and '''B''' and to '''A''' and '''B'''.
+
|| Point to the traces of '''A''' and '''B''' and to '''A''' and '''B'''.
−
| | Note the locus of points traced by '''A''' and '''B'''.
+
|| Note the locus of points traced by points '''A''' and '''B'''.
  
−
These traced points are all equidistant from '''F1''' and '''F2''' (the foci).
+
These traced points are all equidistant from points '''F1''' and '''F2''', the foci.
  
−
They lie on ellipses for which '''F1''' and '''F2''' are foci.
+
They lie on ellipses for which points ''F1''' and '''F2''' are foci.
 
|-
 
|-
−
| | Right-click on '''sliders a''' and '''k''' and uncheck '''Animation On''' option.
+
|| Right-click on '''sliders a''' >> '''k''' >> uncheck '''Animation On''' option.
−
| | Right-click on '''sliders a''' and '''k''' and uncheck '''Animation On''' option.
+
|| Right-click on '''sliders a''' and '''k''' and uncheck '''Animation On''' option.
 
|-
 
|-
−
| | Drag '''sliders a''' and '''k'''.
+
|| Drag '''sliders a''' and '''k'''.
−
| | Drag '''sliders a''' and '''k''' to different values to see more traces of ellipses.
+
|| Drag '''sliders a''' and '''k''' to different values to see more traces of ellipses.
 
|-
 
|-
−
| | Set '''slider k''' between 9 and 10 and '''slider a''' between 5 and 6.
+
|| Set '''slider k''' between 9 and 10 >> '''slider a''' between 5 and 6.
−
| | Set '''slider k''' between 9 and 10 and '''slider a''' between 5 and 6.
+
|| Set '''slider k''' between 9 and 10 and '''slider a''' between 5 and 6.
 
|-
 
|-
−
| | Point to length labels next to '''AF1''' and '''AF2''' and '''sliders a''' and '''k'''.
+
|| Point to length labels next to '''AF1''' and '''AF2''' and '''sliders a''' and '''k'''.
−
| | Note that for a given value of '''k''', as '''a''' changes, lengths of '''Segments AF1''' and '''AF2''' change.
+
|| Note that for a given value of '''k''', as '''a''' changes, lengths of '''Segments AF1''' and '''AF2''' change.
  
 
But their sum remains equal to the value of '''k'''.
 
But their sum remains equal to the value of '''k'''.
 
|-
 
|-
−
| | Point to '''BF1''' and '''BF2''' and '''sliders a''' and '''k'''.
+
|| Point to '''BF1''' and '''BF2''' and '''sliders a''' and '''k'''.
−
| | Note the same fact for Segments '''BF1''' and '''BF2'''.
+
|| Note the same fact for Segments '''BF1''' and '''BF2'''.
 
|-
 
|-
−
| | Click in and move '''Graphics''' view slightly to erase the trace points.
+
|| Click in and move '''Graphics''' view slightly to erase the trace points.
−
| | Click in and move '''Graphics''' view slightly to erase the trace points.
+
|| Click in and move '''Graphics''' view slightly to erase the trace points.
 
|-
 
|-
−
| | Click on the '''Move''' tool and move '''F1''' and '''F2''' to different positions in '''Graphics View'''.
+
|| Click on the '''Move''' tool >> move points '''F1''' and '''F2''' to different positions in '''Graphics View'''.
−
| | Click on '''Move''' tool and move '''F1''' and '''F2''' to different positions in '''Graphics View'''.
+
|| Click on '''Move''' tool and move points '''F1''' and '''F2''' to different positions in '''Graphics View'''.
 
|-
 
|-
−
| | Point to '''sliders a''' and '''k'''.
+
|| Point to '''sliders a''' and '''k'''.
−
| | Values can be changed on '''sliders a''' and '''k''' to see various ellipses.
+
|| Values can be changed on '''sliders a''' and '''k''' to see various ellipses.
 
|-
 
|-
−
| | Open a new '''GeoGebra''' window.
+
|| Open a new '''GeoGebra''' window.
−
| | Let us look at the equations of ellipses in a new '''GeoGebra''' window.
+
|| Let us look at the equations of ellipses in a new '''GeoGebra''' window.
 
|-
 
|-
−
| | In '''input bar''', type '''(x-h)^2/a^2+(y-k)^2/b^2=1''' >> press '''Enter'''.
+
|| In '''input bar''', type '''(x-h)^2/a^2+(y-k)^2/b^2=1''' >> press '''Enter'''.
−
| | In the '''input bar''', type the following line describing the sum of two fractions equal to 1.
+
|| In the '''input bar''', type the following line describing the sum of two fractions equal to 1.
  
 
To type the '''caret symbol''', hold '''Shift''' key down and press 6.
 
To type the '''caret symbol''', hold '''Shift''' key down and press 6.
  
−
For the 1<sup>st</sup> fraction, type the numerator as '''x minus h in parentheses caret 2'''.
+
For the 1<sup>st</sup> fraction, type the numerator as '''x minus h''' in parentheses '''caret 2'''.
  
 
Then type '''division slash'''.
 
Then type '''division slash'''.
Line 297: Line 287:
 
Now, type the denominator of the 1<sup>st</sup> fraction as '''a caret 2''' followed by '''plus'''.
 
Now, type the denominator of the 1<sup>st</sup> fraction as '''a caret 2''' followed by '''plus'''.
  
−
For the 2<sup>nd</sup> fraction, type the numerator as '''y minus k in parentheses caret 2'''.
+
For the 2<sup>nd</sup> fraction, type the numerator as '''y minus k''', in parentheses '''caret 2'''.
  
 
Then type '''division slash'''.
 
Then type '''division slash'''.
Line 305: Line 295:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Point to the pop-up window.
+
|| Point to the pop-up window.
−
| | A pop-up window asks if you want to create '''sliders''' for '''a, b, h''' and '''k'''.
+
|| A pop-up window asks if you want to create '''sliders''' for '''a, b, h''' and '''k'''.
 
|-
 
|-
−
| | Click on '''Create Sliders'''.
+
|| Click on '''Create Sliders'''.
−
| | Click on '''Create Sliders'''.
+
|| Click on '''Create Sliders'''.
 
|-
 
|-
−
| | Point to sliders '''a, b, h''' and '''k'''.
+
|| Point to sliders '''a, b, h''' and '''k'''.
−
| | This creates number '''sliders''' for '''h, a, k''' and '''b'''.
+
|| This creates number '''sliders''' for '''h, a, k''' and '''b'''.
  
 
By default, they go from minus 5 to 5 and are set at 1.
 
By default, they go from minus 5 to 5 and are set at 1.
Line 318: Line 308:
 
You can double-click on the '''sliders''' to see their properties.
 
You can double-click on the '''sliders''' to see their properties.
 
|-
 
|-
−
| | Point to circle '''c''' and center ''(h, k)''' in '''Graphics''' view.
+
|| Point to circle '''c''' and center '''(h, k)''' in '''Graphics''' view.
−
| | A circle '''c''', a special case of an ellipse, appears in '''Graphics''' view.
+
|| A circle '''c''', a special case of an ellipse, appears in '''Graphics''' view.
  
−
Center '''h comma k''' is at '''1 comma 1''' and radius is 1 unit.
+
Centre '''h comma k''' is at '''1 comma 1''' and radius is 1 unit.
 
|-
 
|-
−
| | Point to equation for circle '''c''' in '''Algebra''' view.
+
|| Point to equation for circle '''c''' in '''Algebra''' view.
−
| | In '''Algebra''' view, note the equation for circle '''c'''.
+
|| In '''Algebra''' view, note the equation for circle '''c'''.
 
|-
 
|-
−
| | Drag boundary to see it properly.
+
|| Drag boundary to see it properly.
−
| | Drag the boundary to see it properly.
+
|| Drag the boundary to see it properly.
 
|-
 
|-
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
+
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
+
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
 
|-
 
|-
−
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view.
+
|| Click on '''Move Graphics View''' tool >> drag '''Graphics''' view.
−
| | Click on '''Move Graphics View''' tool and drag '''Graphics''' view.
+
|| Click on '''Move Graphics View''' tool and drag '''Graphics''' view.
 
|-
 
|-
−
| | Point to the equations in '''Algebra''' view as you change '''a''' and '''b''' on '''sliders'''.
+
|| Point to the equations in '''Algebra''' view as you change '''a''' and '''b''' on '''sliders'''.
−
| | Keep track of the equations in '''Algebra''' view as you change '''a''' and '''b''' on the '''sliders'''.
+
|| Keep track of the equations in '''Algebra''' view as you change '''a''' and '''b''' on the '''sliders'''.
 
|-
 
|-
−
| | Place '''cursor''' on equation '''c''' in '''Algebra''' view.
+
|| Place '''cursor''' on equation '''c''' in '''Algebra''' view.
−
| | Place the '''cursor''' on equation '''c''' in '''Algebra''' view.
+
|| Place the '''cursor''' on equation '''c''' in '''Algebra''' view.
 
|-
 
|-
−
| | Point to '''slider a'''.
+
|| Point to '''slider a'''.
  
 
Drag '''slider a''', leave between -2 and -3.
 
Drag '''slider a''', leave between -2 and -3.
−
| | '''a''' is associated with the '''x minus h squared''' component of the equation.
+
|| '''a''' is associated with the '''x minus h squared''' component of the equation.
  
 
Observe how '''a''' controls the horizontal axis of the ellipse.
 
Observe how '''a''' controls the horizontal axis of the ellipse.
 
|-
 
|-
−
| | Point to '''slider b'''.
+
|| Point to '''slider b'''.
  
 
Drag '''slider b''', leave at 5.
 
Drag '''slider b''', leave at 5.
−
| | Associated with the '''y minus k squared''' component is '''b'''.
+
|| Associated with the '''y minus k squared''' component is '''b'''.
  
 
Observe how '''b''' controls the vertical axis of the ellipse.
 
Observe how '''b''' controls the vertical axis of the ellipse.
 
|-
 
|-
−
| | Drag '''slider a''' to 2 and '''b''' to 1.
+
|| Drag '''slider a''' to 2 and '''b''' to 1.
−
| | Drag '''slider a''' to 2 and '''b''' to 1.
+
|| Drag '''slider a''' to 2 and '''b''' to 1.
 
|-
 
|-
−
| | Point to major axis of the ellipse and along the '''x axis'''.
+
|| Point to major axis of the ellipse and along the '''x axis'''.
  
 
Point to equation of the ellipse.
 
Point to equation of the ellipse.
−
| | When '''a''' is greater than '''b''', the major axis of the ellipse is horizontal.
+
|| When '''a''' is greater than '''b''', the major axis of the ellipse is horizontal.
  
 
Note the equation of the ellipse.
 
Note the equation of the ellipse.
 
|-
 
|-
−
| | Drag boundary to see the equation properly.
+
|| Type '''Focus(c)''' in the '''input bar''' and press '''Enter'''.
−
| | Drag the boundary to see it properly.
+
|| In the '''input bar''', type '''Focus c''' in parentheses and press '''Enter'''.
 
|-
 
|-
−
| | Type '''Focus(c)''' in the '''input bar''' and press '''Enter'''.
+
|| Point to '''A''' and '''B''' mapped in '''Graphics''' view and their '''co-ordinates''' in '''Algebra''' view.
−
| | In the '''input bar''', type '''Focus c in parentheses''' and press '''Enter'''.
+
|| Two foci, '''A''' and '''B''', are mapped in '''Graphics''' view and their '''coordinates''' appear in '''Algebra''' view.
 
|-
 
|-
−
| | Point to '''A''' and '''B''' mapped in '''Graphics''' view and their '''co-ordinates''' in '''Algebra''' view.
+
|| Type '''Center(c)''' in the '''input bar''' >> press '''Enter'''.
−
| | Two foci, '''A''' and '''B''', are mapped in '''Graphics''' view and their '''coordinates''' appear in '''Algebra''' view.
+
|| In the '''input bar''', type '''Center c''' in parentheses and press '''Enter.'''
 
|-
 
|-
−
| | Type '''Center(c)''' in the '''input bar''' and press '''Enter'''.
+
|| Point to point '''C''' in '''Graphics''' view and its '''co-ordinates''' in '''Algebra''' view.
−
| | In the '''input bar''', type '''Center c in parentheses''' and press '''Enter.'''
+
|| Center '''C''' appears in '''Graphics''' view and its '''co-ordinates''' appear in '''Algebra''' view.
 
|-
 
|-
−
| | Point to point '''C''' in '''Graphics''' view and its '''co-ordinates''' in '''Algebra''' view.
+
|| Type '''Vertex(c)''' in the '''input bar''' >> press '''Enter'''.
−
| | Center '''C''' appears in '''Graphics''' view and its '''co-ordinates''' appear in '''Algebra''' view.
+
|| In the '''input bar''', type '''Vertex c''' in parentheses and press '''Enter'''.
−
|-
+
−
| | Type '''Vertex(c)''' in the '''input bar''' >> press '''Enter'''.
+
−
| | In the '''input bar''', type '''Vertex c in parentheses''' and press '''Enter'''.
+
  
 
Vertices '''D''' and '''E''' appear at the ends of the major axis.
 
Vertices '''D''' and '''E''' appear at the ends of the major axis.
Line 387: Line 374:
 
Co-vertices '''F''' and '''G''' appear at the ends of the minor axis.
 
Co-vertices '''F''' and '''G''' appear at the ends of the minor axis.
 
|-
 
|-
−
| | Click on '''Text''' tool under '''Slider''' tool and click in '''Graphics''' view.
+
|| under '''Slider''' tool >> Click on '''Text''' tool >> click in '''Graphics''' view.
−
| | Under '''Slider''', click on '''Text''' tool and click in '''Graphics''' view.
+
|| Under '''Slider''', click on '''Text''' tool and click in '''Graphics''' view.
 
|-
 
|-
−
| | Point to the '''text-box'''.
+
|| Point to the '''text-box'''.
−
| | A '''text-box''' opens up.
+
|| A '''text-box''' opens up.
 
|-
 
|-
−
| | '''Slide Number 6'''
+
|| '''Slide Number 6'''
  
 
'''Text box for ellipse c'''
 
'''Text box for ellipse c'''
Line 406: Line 393:
  
 
'''latus rectum''' = 1
 
'''latus rectum''' = 1
−
| | '''Text box for ellipse c'''
+
|| In the '''Edit''' field, type the following text.
−
 
+
−
In the '''Edit''' field, type the following text.
+
  
 
Press '''Enter''' after each line to go to the next line.
 
Press '''Enter''' after each line to go to the next line.
Line 414: Line 399:
 
Click '''OK'''.
 
Click '''OK'''.
 
|-
 
|-
−
| |
+
||
−
| | Refer to additional material provided with this tutorial for these calculations.
+
|| Refer to additional material provided with this tutorial for these calculations.
 +
|-
 +
|| Leave '''slider a''' at 2, drag '''slider b''' to 3.
 +
|| Leave '''slider a''' at 2, drag '''slider b''' to 3.
 
|-
 
|-
−
| | Leave '''slider a''' at 2, drag '''slider b''' to 3.
+
|| Point to the ellipse '''c''' and major and minor axes.
−
| | Leave '''slider a''' at 2, drag '''slider b''' to 3.
+
|| Note the effects on the shape of ellipse '''c''' and the change in directions of major and minor axes.
 
|-
 
|-
−
| | Point to the ellipse '''c''' and major and minor axes.
+
|| Point to the equation in '''Algebra''' view.
−
| | Note the effects on the shape of ellipse '''c''' and the change in directions of major and minor axes.
+
|| Note also the change in the equation in '''Algebra''' view.
 
|-
 
|-
−
| | Point to the equation in '''Algebra''' view.
+
||Drag the boundary.
−
| | Note also the change in the equation in '''Algebra''' view.
+
||Drag boundary to see it properly.
 
|-
 
|-
−
| |
+
||Point to the ellipse.
−
| | Calculate eccentricity and length of latus recti, major and minor axes for this ellipse.
+
|| Calculate eccentricity and length of latus recti, major and minor axes for this ellipse.
 
|-
 
|-
−
| | Show everything unchecked in '''Algebra''' view.
+
|| Show everything unchecked in '''Algebra''' view.
−
| | In '''Algebra''' view, uncheck ellipse '''c''' and all points and text generated for it to hide them.
+
|| In '''Algebra''' view, uncheck ellipse '''c''' and all points and text generated for it to hide them.
 
|-
 
|-
−
| | Show screenshots of ellipse '''d''' for '''a = 2, b =1''' and '''a =2, b = 3'''.
+
|| Show screenshots of ellipse '''d''' for '''a = 2, b =1''' and '''a =2, b = 3'''.
−
| | Follow the earlier steps to construct ellipse '''d''' for these two conditions.
+
|| Follow the earlier steps to construct ellipse '''d''' for these two conditions.
 
|-
 
|-
−
| |
+
||
−
| | Let us summarize.
+
|| Let us summarize.
 
|-
 
|-
−
| | '''Slide Number 8'''
+
|| '''Slide Number 7'''
  
 
'''Summary'''
 
'''Summary'''
−
| | In this tutorial, we have learnt how to:
+
|| In this tutorial, we have learnt how to:
  
 
Use '''GeoGebra''' to construct an ellipse
 
Use '''GeoGebra''' to construct an ellipse
Line 447: Line 435:
 
Look at standard equations and parts of an ellipse
 
Look at standard equations and parts of an ellipse
 
|-
 
|-
−
| | '''Slide Number 9'''
+
|| '''Slide Number 8'''
  
 
'''Assignment'''
 
'''Assignment'''
Line 460: Line 448:
  
 
Calculate eccentricity and length of latus recti, major and minor axes.
 
Calculate eccentricity and length of latus recti, major and minor axes.
−
| | As an '''assignment''',
+
|| As an '''assignment''',
  
 
Construct ellipses with the following foci and vertices.
 
Construct ellipses with the following foci and vertices.
Line 466: Line 454:
 
Find all these values.
 
Find all these values.
 
|-
 
|-
−
| | '''Slide Number 10'''
+
|| '''Slide Number 9'''
  
 
'''Assignment'''
 
'''Assignment'''
Line 477: Line 465:
  
 
'''36x<sup>2</sup> + 4y<sup>2</sup> = 144'''
 
'''36x<sup>2</sup> + 4y<sup>2</sup> = 144'''
−
| | Find all these values for these ellipses.
+
|| Find all these values for these ellipses.
 
|-
 
|-
−
| | '''Slide Number 11'''
+
|| '''Slide Number 10'''
  
 
'''About Spoken Tutorial project'''
 
'''About Spoken Tutorial project'''
−
| | The video at the following link summarizes the Spoken Tutorial project.
+
|| The video at the following link summarizes the Spoken Tutorial project.
  
 
Please download and watch it.
 
Please download and watch it.
 
|-
 
|-
−
| | '''Slide Number 12'''
+
|| '''Slide Number 11'''
  
 
'''Spoken Tutorial workshops'''
 
'''Spoken Tutorial workshops'''
−
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.
+
|| The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.
  
 
For more details, please write to us.
 
For more details, please write to us.
 
|-
 
|-
−
| | '''Slide Number 13'''
+
|| '''Slide Number 12'''
  
 
'''Forum for specific questions:'''
 
'''Forum for specific questions:'''
Line 507: Line 495:
 
Someone from our team will answer them.
 
Someone from our team will answer them.
  
−
| | Please post your timed queries on this forum.
+
|| Please post your timed queries on this forum.
 
|-
 
|-
−
| | '''Slide Number 14'''
+
|| '''Slide Number 13'''
  
 
'''Acknowledgement'''
 
'''Acknowledgement'''
−
| | '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.
+
|| '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.
  
 
More information on this mission is available at this link.
 
More information on this mission is available at this link.
 
|-
 
|-
−
| |
+
||
−
| | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.
+
|| This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.
  
 
Thank you for joining.
 
Thank you for joining.
 
|-
 
|-
 
|}
 
|}

Latest revision as of 12:18, 11 September 2019

Visual Cue Narration
Slide Number 1

Title Slide

Welcome to this tutorial on Conic Sections - Ellipse
Slide Number 2

Learning Objectives

In this tutorial, we will learn,

Standard equations and parts of an ellipse

To use GeoGebra to construct an ellipse

Slide Number 3

System Requirement

Here, I am using:

Ubuntu Linux OS version 14.04

GeoGebra 5.0.388.0-d

Slide Number 4

Pre-requisites

www.spoken-tutorial.org

To follow this tutorial, you should be familiar with

GeoGebra interface

Conic sections in geometry

For relevant tutorials, please visit our website.

Slide Number 5

Ellipse

An ellipse, is the locus of points whose sum of distances from two fixed points is constant.

These fixed points are called the foci.

Point to the center O, foci F1 and F2.

Point to vertices A and B at the ends of the major axis AB.

Point to C and D at the ends of the minor axis CD.

Point to 2 latus recti passing through foci and the two axes.

Ellipse

An ellipse is the locus of points whose sum of distances from two fixed points is constant.

These fixed points are called the foci.

Observe the centre O, foci F1 and F2.

Vertices A and B are at the ends of the major axis AB.

Co-vertices C and D are at the ends of the minor axis CD.

Two latus recti pass through the foci.

Axes lengths 2a and 2b and distance between the foci 2c are shown in the figure.

Be careful to distinguish length, from letters used for sliders, circles and ellipses.
Show the GeoGebra window. Let us construct an ellipse in GeoGebra.

I have already opened the GeoGebra interface.

Click on Point tool >> click twice in Graphics view. Click on Point tool and click twice in Graphics view.
Point to A and B. This creates two points A and B, which will be the foci of our ellipse.
Right-click on A >> choose Rename option. Right-click on A and choose the Rename option.
Type F1 in the New Name field >> click OK button. In the New Name field, type F1 and click OK.
Point to the focus F1. This will be one of our foci, F1.
Point to B. Let us rename B as F2.
Click on Slider tool >> click in Graphics view. Click on Slider tool and click in Graphics view.
Point to the Slider dialog box in Graphics view. Slider dialog box appears in Graphics view.
Stay with the default Number selection >> in the Name field, type k. Stay with the default Number selection and in the Name field, type k.
Set Min value as 0, Max value as 10, increment as 0.1.

Click OK.

Set Min value as 0, Max value as 10, increment as 0.1.

Click OK.

Point to slider k. This creates a number slider named k.
Drag slider k. Slider k can be changed from 0 to 10.

k will be the sum of the distances of any point on the ellipse from the foci F1 and F2.

point to a. We will create another number slider named a.
Set Min value as 0, Max value as 10, increment as 0.1.

Click OK.

Its Min value is 0, Max value is 10, increment is 0.1."
Click on Circle with Center and Radius tool >> click on F1. Click on Circle with Center and Radius tool and click on F1.
Point to the text box; type a >> click OK. A text-box appears. In the Radius name field, type a and click OK.
Point to the circle with centre F1 >> radius a. A circle c with centre F1 and radius a appears.
Drag slider a to 2 >> slider k to 5. Drag slider a to 2 and slider k to 5.
Click on Circle with Center and Radius tool >> click on F2. Click again on Circle with Center and Radius tool and click on F2.
Point to the text box; type k-a and click OK. In the text box that appears, type k minus a and click OK.
Point to circle d with center F2 >> radius k-a in Graphics view. A circle d with center F2 and radius k minus a appears in Graphics view.
Under Move Graphics View, click on Zoom Out tool and in Graphics view. Under Move Graphics View, click on Zoom Out and in Graphics view.
Click on Move Graphics View >> drag Graphics view. Click on Move Graphics View and drag Graphics view.
Click on Intersect tool under Point tool.

Click on the intersections of the two circles.

Under Point, click on Intersect tool.

Click on the two circles c and d.

Point to points A and B. This creates points A and B.
Click on Segment tool under Line tool. Under Line, click on Segment tool.
Click on points F1 >> A to join. Click on points F1 and A to join them.
Click on points A >> F2 to join. Next, click on points A and F2 to join them.
Click on Move tool. Click on Move.
Double click on Segment AF1. Double click on Segment AF1.
Click Object Properties to open the Preferences dialog box. Click on Object Properties to open the Preferences dialog box.
Point to Segment AF1 highlighted in the left panel. Segment AF1 is already highlighted in the left panel.
Holding Ctrl key down, click >> highlight Segment AF2. Holding Ctrl key down, click and highlight Segment AF2 as well.
Under Basic tab >> select Show Label. Under the Basic tab, make sure that Show Label is selected.
Pull down the drop down menu next to the Show Label check box.

Select Name and Value.

Pull down the drop down menu next to the Show Label check box.

Select Name and Value.

Under Color tab >> select Red. Under the Color tab, select red.
Under Style tab >> choose dashed line style. Under the Style tab, choose dashed line style.
Close the Preferences dialog box. Close the Preferences dialog box.
Follow earlier steps to draw Segments BF1 and BF2.

Make them dashed and blue.

Draw Segments BF1 and BF2.

Make them dashed and blue.

Point to Move tool.

If not highlighted, click on it.

Move the labels so you can see them properly.

Make sure that the Move tool is highlighted.

Move the labels so you can see them properly.

Point to the length labels next to AF1, AF2, BF1, BF2 and slider k. Note that the sum of the segment lengths from both foci to each intersection point is equal to k.
Right-click on A and B >> check Trace On option. Right-click on A and B and check Trace On option.
Uncheck circles c and d in Algebra view. In Algebra view, uncheck circles c and d to hide the circles.
Right-click on slider a >> check Animation On option. Right-click on slider a and check Animation On option.
Right-click on slider k >> check Animation On option. Next, right-click on slider k and check Animation On option.
Point to the traces of A and B and to A and B. Note the locus of points traced by points A and B.

These traced points are all equidistant from points F1 and F2, the foci.

They lie on ellipses for which points F1' and F2 are foci.

Right-click on sliders a >> k >> uncheck Animation On option. Right-click on sliders a and k and uncheck Animation On option.
Drag sliders a and k. Drag sliders a and k to different values to see more traces of ellipses.
Set slider k between 9 and 10 >> slider a between 5 and 6. Set slider k between 9 and 10 and slider a between 5 and 6.
Point to length labels next to AF1 and AF2 and sliders a and k. Note that for a given value of k, as a changes, lengths of Segments AF1 and AF2 change.

But their sum remains equal to the value of k.

Point to BF1 and BF2 and sliders a and k. Note the same fact for Segments BF1 and BF2.
Click in and move Graphics view slightly to erase the trace points. Click in and move Graphics view slightly to erase the trace points.
Click on the Move tool >> move points F1 and F2 to different positions in Graphics View. Click on Move tool and move points F1 and F2 to different positions in Graphics View.
Point to sliders a and k. Values can be changed on sliders a and k to see various ellipses.
Open a new GeoGebra window. Let us look at the equations of ellipses in a new GeoGebra window.
In input bar, type (x-h)^2/a^2+(y-k)^2/b^2=1 >> press Enter. In the input bar, type the following line describing the sum of two fractions equal to 1.

To type the caret symbol, hold Shift key down and press 6.

For the 1st fraction, type the numerator as x minus h in parentheses caret 2.

Then type division slash.

Now, type the denominator of the 1st fraction as a caret 2 followed by plus.

For the 2nd fraction, type the numerator as y minus k, in parentheses caret 2.

Then type division slash.

Now, type the denominator of the 2nd fraction as b caret 2 followed by equals sign 1.

Press Enter.

Point to the pop-up window. A pop-up window asks if you want to create sliders for a, b, h and k.
Click on Create Sliders. Click on Create Sliders.
Point to sliders a, b, h and k. This creates number sliders for h, a, k and b.

By default, they go from minus 5 to 5 and are set at 1.

You can double-click on the sliders to see their properties.

Point to circle c and center (h, k) in Graphics view. A circle c, a special case of an ellipse, appears in Graphics view.

Centre h comma k is at 1 comma 1 and radius is 1 unit.

Point to equation for circle c in Algebra view. In Algebra view, note the equation for circle c.
Drag boundary to see it properly. Drag the boundary to see it properly.
Under Move Graphics View, click on Zoom Out tool and in Graphics view. Under Move Graphics View, click on Zoom Out tool and in Graphics view.
Click on Move Graphics View tool >> drag Graphics view. Click on Move Graphics View tool and drag Graphics view.
Point to the equations in Algebra view as you change a and b on sliders. Keep track of the equations in Algebra view as you change a and b on the sliders.
Place cursor on equation c in Algebra view. Place the cursor on equation c in Algebra view.
Point to slider a.

Drag slider a, leave between -2 and -3.

a is associated with the x minus h squared component of the equation.

Observe how a controls the horizontal axis of the ellipse.

Point to slider b.

Drag slider b, leave at 5.

Associated with the y minus k squared component is b.

Observe how b controls the vertical axis of the ellipse.

Drag slider a to 2 and b to 1. Drag slider a to 2 and b to 1.
Point to major axis of the ellipse and along the x axis.

Point to equation of the ellipse.

When a is greater than b, the major axis of the ellipse is horizontal.

Note the equation of the ellipse.

Type Focus(c) in the input bar and press Enter. In the input bar, type Focus c in parentheses and press Enter.
Point to A and B mapped in Graphics view and their co-ordinates in Algebra view. Two foci, A and B, are mapped in Graphics view and their coordinates appear in Algebra view.
Type Center(c) in the input bar >> press Enter. In the input bar, type Center c in parentheses and press Enter.
Point to point C in Graphics view and its co-ordinates in Algebra view. Center C appears in Graphics view and its co-ordinates appear in Algebra view.
Type Vertex(c) in the input bar >> press Enter. In the input bar, type Vertex c in parentheses and press Enter.

Vertices D and E appear at the ends of the major axis.

Co-vertices F and G appear at the ends of the minor axis.

under Slider tool >> Click on Text tool >> click in Graphics view. Under Slider, click on Text tool and click in Graphics view.
Point to the text-box. A text-box opens up.
Slide Number 6

Text box for ellipse c

Major axis 2a = 4

Minor axis 2b = 2

c = 1.732

e = 0.866

latus rectum = 1

In the Edit field, type the following text.

Press Enter after each line to go to the next line.

Click OK.

Refer to additional material provided with this tutorial for these calculations.
Leave slider a at 2, drag slider b to 3. Leave slider a at 2, drag slider b to 3.
Point to the ellipse c and major and minor axes. Note the effects on the shape of ellipse c and the change in directions of major and minor axes.
Point to the equation in Algebra view. Note also the change in the equation in Algebra view.
Drag the boundary. Drag boundary to see it properly.
Point to the ellipse. Calculate eccentricity and length of latus recti, major and minor axes for this ellipse.
Show everything unchecked in Algebra view. In Algebra view, uncheck ellipse c and all points and text generated for it to hide them.
Show screenshots of ellipse d for a = 2, b =1 and a =2, b = 3. Follow the earlier steps to construct ellipse d for these two conditions.
Let us summarize.
Slide Number 7

Summary

In this tutorial, we have learnt how to:

Use GeoGebra to construct an ellipse

Look at standard equations and parts of an ellipse

Slide Number 8

Assignment

Construct ellipses with:

Foci (± 4, 0) and vertices (± 5, 0)

Foci (0, ± 5) and vertices (0, ± 13)

Find their centres and equations.

Calculate eccentricity and length of latus recti, major and minor axes.

As an assignment,

Construct ellipses with the following foci and vertices.

Find all these values.

Slide Number 9

Assignment

Find the coordinates of the foci, vertices and co-vertices.

Eccentricity and length of major, minor axes and atus rectum for these ellipses:

x2/4 + y2/25 = 1

36x2 + 4y2 = 144

Find all these values for these ellipses.
Slide Number 10

About Spoken Tutorial project

The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.

Slide Number 11

Spoken Tutorial workshops

The Spoken Tutorial Project team conducts workshops and gives certificates.

For more details, please write to us.

Slide Number 12

Forum for specific questions:

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

Please post your timed queries on this forum.
Slide Number 13

Acknowledgement

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This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Madhurig, Snehalathak, Vidhya

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