PoojaMoolya: Created page with "{|border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to this tutorial on '''Conic Sections - Ellipse''' |- || 00:06 || In this tutorial, we will learn, |- || 0..."

2020-10-21T07:25:45Z

Created page with "{|border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to this tutorial on '''Conic Sections - Ellipse''' |- || 00:06 || In this tutorial, we will learn, |- || 0..."

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{|border=1
|| '''Time'''
|| '''Narration'''

|-
|| 00:01
|| Welcome to this tutorial on '''Conic Sections - Ellipse'''
|-
|| 00:06
|| In this tutorial, we will learn,

|-
|| 00:10
|| '''Standard equations''' and parts of an ellipse

|-
|| 00:14
|| To use '''GeoGebra''' to construct an ellipse
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|| 00:18
|| Here, I am using:

'''Ubuntu Linux ''' OS version 14.04

'''GeoGebra 5.0.388.0 hyphen d'''
|-
|| 00:31
|| To follow this '''tutorial''', you should be familiar with

'''GeoGebra''' interface, Conic sections in geometry

|-
|| 00:39
|| For relevant tutorials, please visit our website.
|-
|| 00:43
||'''Ellipse'''

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

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|| 00:55
|| These fixed points are called the foci.

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|| 01:00
|| Observe the centre '''O''', foci '''F1''' and '''F2'''.

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|| 01:08
|| Vertices '''A''' and '''B''' are at the ends of the major axis '''AB'''.

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|| 01:15
|| '''Co-vertices C''' and '''D''' are at the ends of the minor axis '''CD'''.

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|| 01:23
|| Two latus recti pass through the foci.

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|| 01:27
|| Axes lengths '''2a''' and '''2b''' and distance between the foci '''2c''' are shown in the figure.
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||01:37
|| Be careful to distinguish length, from letters used for '''sliders''', circles and ellipses.
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|| 01:45
|| Let us construct an ellipse in '''GeoGebra'''.

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|| 01:49
|| I have already opened the '''GeoGebra''' interface.
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|| 01:54
|| Click on '''Point''' tool and click twice in '''Graphics''' view.
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|| 02:03
|| This creates two points '''A''' and '''B''', which will be the foci of our ellipse.
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|| 02:10
|| Right-click on '''A''' and choose the '''Rename''' option.
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|| 02:16
|| In the '''New Name''' field, type '''F1''' and click '''OK'''.
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|| 02:23
|| This will be one of our foci, '''F1'''.
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|| 02:28
|| Let us rename '''B''' as '''F2'''.
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|| 02:32
|| Click on '''Slider''' tool and click in '''Graphics''' view.
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|| 02:40
|| '''Slider dialog box''' appears in '''Graphics''' view.
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|| 02:45
|| Stay with the default '''Number''' selection and in the '''Name''' field, type '''k'''.
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|| 02:53
|| Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1.

Click '''OK'''.
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|| 03:07
|| This creates a number '''slider''' named '''k'''.
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|| 03:12
|| '''Slider k''' can be changed from 0 to 10.

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|| 03:17
|| '''k''' will be the sum of the distances of any point on the ellipse from the foci '''F1''' and '''F2'''.
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||03:26
|| We will create another '''number slider''' named '''a'''.

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|| 03:31
|| Its '''Min''' value is 0, '''Max''' value is 10, '''increment''' is 0.1."
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|| 03:39
|| Click on '''Circle with Center and Radius''' tool and click on '''F1'''.
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||03:49
|| A '''textbox''' appears. In the '''Radius''' name field, type '''a''' and click '''OK'''.
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|| 04:00
|| A circle '''c''' with centre '''F1''' and radius '''a''' appears.
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|| 04:08
|| Drag '''slider a''' to 2 and '''slider k''' to 5.
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|| 04:16
|| Click again on '''Circle with Center and Radius''' tool and click on '''F2'''.
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|| 04:26
|| In the '''text box''' that appears, type '''k minus a''' and click '''OK'''.
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|| 04:35
|| A circle '''d''' with center '''F2''' and radius '''k minus a''' appears in '''Graphics''' view.
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|| 04:44
|| Under '''Move Graphics View''', click on '''Zoom Out''' and in '''Graphics''' view.
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|| 04:53
|| Click on '''Move Graphics View''' and drag '''Graphics''' view.
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|| 05:01
|| Under '''Point''', click on '''Intersect''' tool.

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|| 05:06
|| Click on the two circles '''c''' and '''d'''.
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|| 05:12
|| This creates points '''A''' and '''B'''.
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|| 05:16
|| Under '''Line''', click on '''Segment''' tool.
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|| 05:21
|| Click on points '''F1''' and '''A''' to join them.
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|| 05:27
|| Next, click on points '''A''' and '''F2''' to join them.
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|| 05:33
|| Click on '''Move'''.
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|| 05:36
|| Double click on '''Segment AF1'''.
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|| 05:40
|| Click on '''Object Properties''' to open the '''Preferences''' dialog box.
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|| 05:45
|| Segment '''AF1''' is already highlighted in the left panel.
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|| 05:51
|| Holding '''Ctrl''' key down, click and highlight Segment '''AF2''' as well.
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|| 05:58
|| Under the '''Basic''' tab, make sure that '''Show Label''' is selected.
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|| 06:04
|| Pull down the '''drop down menu''' next to the '''Show Label''' check box.

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|| 06:09
|| Select '''Name and Value'''.
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|| 06:12
|| Under the '''Color''' tab, select red.
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|| 06:16
|| Under the '''Style''' tab, choose '''dashed line style'''.
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|| 06:23
|| Close the '''Preferences''' dialog box.
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|| 06:26
|| Draw Segments '''BF1''' and '''BF2'''.

Make them '''dashed''' and blue.
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|| 06:31
|| Make sure that the '''Move''' tool is highlighted.

|-
|| 06:39
|| Move the labels so you can see them properly.
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|| 06:45
|| Note that the sum of the segment lengths from both foci to each intersection point is equal to '''k'''.
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|| 06:55
|| Right-click on '''A''' and '''B''' and check '''Trace On''' option.
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|| 07:04
|| In '''Algebra''' view, uncheck circles '''c''' and '''d''' to hide the circles.
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|| 07:10
|| Right-click on '''slider a''' and check '''Animation On''' option.
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|| 07:18
|| Next, right-click on '''slider k''' and check '''Animation On''' option.
|-
|| 07:26
|| Note the locus of points traced by points '''A''' and '''B'''.

|-
|| 07:31
|| These traced points are all equidistant from points '''F1''' and '''F2''', the foci.

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|| 07:39
|| They lie on ellipses for which points ''F1''' and '''F2''' are foci.
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|| 07:45
|| Right-click on '''sliders a''' and '''k''' and uncheck '''Animation On''' option.
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|| 07:56
|| Drag '''sliders a''' and '''k''' to different values to see more traces of ellipses.
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|| 08:06
|| Set '''slider k''' between 9 and 10 and '''slider a''' between 5 and 6.
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|| 08:15
|| 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'''.
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|| 08:27
|| Note the same fact for Segments '''BF1''' and '''BF2'''.
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|| 08:33
|| Click in and move '''Graphics''' view slightly to erase the trace points.
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|| 08:39
|| Click on '''Move''' tool and move points '''F1''' and '''F2''' to different positions in '''Graphics View'''.
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|| 08:50
|| Values can be changed on '''sliders a''' and '''k''' to see various ellipses.
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|| 08:58
|| Let us look at the equations of ellipses in a new '''GeoGebra''' window.
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|| 09:04
|| In the '''input bar''', type the following line describing the sum of two fractions equal to 1.

|-
|| 09:12
|| To type the '''caret symbol''', hold '''Shift''' key down and press 6.

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|| 09:18
|| For the 1st fraction, type the numerator as '''x minus h''' in parentheses '''caret 2'''.

Then type '''division slash'''.

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|| 09:30
|| Now, type the denominator of the 1st fraction as '''a caret 2''' followed by '''plus'''.

|-
|| 09:37
|| For the 2nd fraction, type the numerator as '''y minus k''', in parentheses '''caret 2'''.

|-
|| 09:46
|| Then type '''division slash'''.

|-
|| 09:49
|| Now, type the denominator of the 2nd fraction as '''b caret 2''' followed by '''equals sign 1'''.

Press '''Enter'''.
|-
|| 10:00
|| A pop-up window asks if you want to create '''sliders''' for '''a, b, h''' and '''k'''.
|-
|| 10:07
|| Click on '''Create Sliders'''.
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|| 10:10
|| This creates number '''sliders''' for '''h, a, k''' and '''b'''.

|-
|| 10:17
|| By default, they go from minus 5 to 5 and are set at 1.

|-
|| 10:23
|| You can double-click on the '''sliders''' to see their properties.
|-
|| 10:27
|| A circle '''c''', a special case of an ellipse, appears in '''Graphics''' view.

|-
|| 10:32
|| Centre '''h comma k''' is at '''1 comma 1''' and radius is 1 unit.
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|| 10:39
|| In '''Algebra''' view, note the equation for circle '''c'''.
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|| 10:44
|| Drag the boundary to see it properly.
|-
|| 10:48
|| Under '''Move Graphics View''', click on '''Zoom Out''' tool and in '''Graphics''' view.
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|| 10:56
|| Click on '''Move Graphics View''' tool and drag '''Graphics''' view.
|-
|| 11:02
|| Keep track of the equations in '''Algebra''' view as you change '''a''' and '''b''' on the '''sliders'''.
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|| 11:09
|| Place the '''cursor''' on equation '''c''' in '''Algebra''' view.
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|| 11:14
|| '''a''' is associated with the '''x minus h squared''' component of the equation.

|-
|| 11:21
|| Observe how '''a''' controls the horizontal axis of the ellipse.
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|| 11:28
|| Associated with the '''y minus k squared''' component is '''b'''.

|-
|| 11:34
|| Observe how '''b''' controls the vertical axis of the ellipse.
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|| 11:41
|| Drag '''slider a''' to 2 and '''b''' to 1.
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|| 11:48
|| When '''a''' is greater than '''b''', the major axis of the ellipse is horizontal.

Note the equation of the ellipse.
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|| 11:58
|| In the '''input bar''', type '''Focus c''' in parentheses and press '''Enter'''.
|-
|| 12:07
|| Two foci, '''A''' and '''B''', are mapped in '''Graphics''' view and their '''coordinates''' appear in '''Algebra''' view.
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|| 12:16
|| In the '''input bar''', type '''Center c''' in parentheses and press '''Enter.'''
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|| 12:25
|| Center '''C''' appears in '''Graphics''' view and its '''co-ordinates''' appear in '''Algebra''' view.
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|| 12:32
|| In the '''input bar''', type '''Vertex c''' in parentheses and press '''Enter'''.

|-
|| 12:41
|| Vertices '''D''' and '''E''' appear at the ends of the major axis.

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|| 12:48
|| Co-vertices '''F''' and '''G''' appear at the ends of the minor axis.
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|| 12:54
|| Under '''Slider''', click on '''Text''' tool and click in '''Graphics''' view.
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||13:02
|| A '''text-box''' opens up.
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|| 13:05
|| In the '''Edit''' field, type the following text.

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|| 13:09
|| Press '''Enter''' after each line to go to the next line.

Click '''OK'''.
|-
||13:16
|| Refer to additional material provided with this tutorial for these calculations.
|-
|| 13:23
|| Leave '''slider a''' at 2, drag '''slider b''' to 3.
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|| 13:29
|| Note the effects on the shape of ellipse '''c''' and the change in directions of major and minor axes.
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|| 13:38
|| Note also the change in the equation in '''Algebra''' view.
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||13:43
||Drag boundary to see it properly.
|-
||13:46
|| Calculate eccentricity and length of latus recti, major and minor axes for this ellipse.
|-
|| 13:54
|| In '''Algebra''' view, uncheck ellipse '''c''' and all points and text generated for it to hide them.
|-
|| 14:08
|| Follow the earlier steps to construct ellipse '''d''' for these two conditions.
|-
||14:15
|| Let us summarize.
|-
||14:17
|| In this tutorial, we have learnt how to:

Use '''GeoGebra''' to construct an ellipse

|-
|| 14:24
|| Look at standard equations and parts of an ellipse
|-
|| 14:28
|| As an '''assignment''',

Construct ellipses with the following foci and vertices. Find all these values.
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|| 14:37
|| Find all these values for these ellipses.
|-
|| 14:42
|| The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.
|-
|| 14:51
|| The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
|| 15:00
|| Please post your timed queries on this forum.
|-
|| 15:04
|| '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
||15:17
|| This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

Applications-of-GeoGebra/C2/Conic-Sections-Ellipse/English-timed - Revision history

PoojaMoolya: Created page with "{|border=1 || '''Time''' || '''Narration''' |- || 00:01 || Welcome to this tutorial on '''Conic Sections - Ellipse''' |- || 00:06 || In this tutorial, we will learn, |- || 0..."