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

2020-10-21T07:26:39Z

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

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

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

Study standard equations and parts of hyperbolae

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||00:13
||Learn how to use '''GeoGebra''' to construct a hyperbola.

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||00:18
||Here I am using:

'''Ubuntu Linux OS''' version 14.04 , '''GeoGebra 5.0.388.0 hyphen d'''
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||00:33
||To follow this tutorial, you should be familiar with

'''GeoGebra''' interface

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||00:40
||Conic Sections in geometry

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||00:43
||For relevant tutorials, please visit our website.
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||00:48
||Hyperbola,

Consider two fixed points '''F1''' and '''F2''' called foci.

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||00:57
||A hyperbola is the locus of points whose difference of distances from these foci is constant.

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||01:07
||In the image, observe that foci of a hyperbola lie along the transverse axis.

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||01:15
||They are equidistant from the center which lies on the conjugate axis.

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||01:22
||'''2b''' is the length of the conjugate axis.

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||01:27
||'''c''' is the distance of each focus from the center.

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||01:33
||The conjugate axis is perpendicular to the transverse axis.

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||01:38
||The hyperbola intersects the transverse axis at the vertices '''A ''' and '''B'''.

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||01:46
||'''a''' is the distance of each vertex from the center.

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||01:52
||The latus recti pass through the foci.

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||01:56
||They are perpendicular to the transverse axis.

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||02:00
||Be careful to distinguish lengths from letters used for '''sliders''', circles and hyperbolae.

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||02:08
||Let us construct a hyperbola in '''GeoGebra'''.

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||02:13
||I have already opened the '''GeoGebra''' interface.
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||02:18
||Click on '''Point''' tool and click twice in '''Graphics''' view.

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||02:26
||This creates two points ''' A''' and '''B''', which will be the foci of our hyperbola.
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||02:33
||Right-click on '''A''' and choose the '''Rename''' option.
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||02:39
||In the '''New Name''' field, type '''F1''' and click '''OK'''.
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||02:45
||This will be one of our foci, '''F1'''.
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||02:49
||Let us rename point '''B''' as '''F2'''.
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||02:53
||Click on '''Slider''' tool and click in '''Graphics '''view.
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||03:00
||A '''Slider dialog-box''' appears in '''Graphics''' view.
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||03:04
||Stay with the default '''Number''' selection.

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||03:08
||In the '''Name''' field, type '''k'''.
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||03:12
||Set '''Min''' value as 0, '''Max''' value as 10, '''increment''' as 0.1, click '''OK'''.
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||03:26
||This creates a number '''slider''' named '''k'''.
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||03:30
||Using this '''slider''', '''k''' can be changed from 0 to 10.
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||03:36
||'''k''' will be the difference of the distances of any point on the hyperbola from the foci, '''F1''' and '''F2'''.
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||03:45
||Drag '''slider k''' to 4.
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||03:49
||We will create another '''number slider''' named '''a'''.
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||03:54
||Its '''Min''' value is 0, '''Max''' value is 25,'''increment''' is 0.1.
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||04:02
||Click on '''Circle with Center and Radius''' tool and click on '''F1'''.
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||04:10
||A '''text-box''' appears; type '''a''' and click '''OK'''.
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||04:17
||Drag '''a''' to a value between 2 and 3.
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||04:23
||A circle '''c''' with center '''F1''' and radius '''a''' appears.
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||04:29
||Drag '''slider a''' to 5.
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||04:33
||Under '''Move Graphics View''', click on '''Zoom Out''' tool.
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||04:39
||Click in '''Graphics''' view.
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||04:42
||Click on '''Move Graphics View''' to move the background as required.
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||04:48
||Click again on '''Circle with Center and Radius''' tool and click on ''' F2'''.
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||04:56
||In the '''text-box''', type '''a minus k''' and click '''OK'''.
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||05:03
||Circle '''d''' with center '''F2''' and radius '''a minus k''' appears.
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||05:10
||Click again on '''Circle with Center and Radius''' tool and click on '''F2'''.
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||05:18
||In the '''text-box''', type '''a plus k''' and click '''OK'''.
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||05:25
||Circle '''e''' with center '''F2''' and radius '''a plus k''' appears.
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||05:32
||Set '''slider k''' between 1 and 2, '''slider a''' between 3 and 4.
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||05:40
||Under '''Point''', click on '''Intersect'''.

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||05:46
||Then click on circles '''c''' and '''d''' and circles '''c''' and '''e'''.

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||05:55
||This creates points '''A''', '''B''', '''C''' and '''D'''.
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||06:05
||Under '''Line''', click on '''Segment''' and click on points '''A''' and '''F1''' to join them.

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||06:15
||Then click on points '''A''' and '''F2''' to join them.
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||06:21
||Similarly, using '''Segment''' tool, join '''B''' and '''F1''' as well as '''B''' and '''F2'''.
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||06:31
||Click on '''Move'''.
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||06:34
||Double click on segment '''AF1''' and click on '''Object Properties'''.
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||06:42
||In the left panel, segment '''AF1''' is already highlighted.
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||06:48
||Holding '''Ctrl''' Key down, click and highlight segments '''AF2, BF1''' and '''BF2'''.
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||06:58
||Under the '''Basic''' tab, make sure '''Show Label''' is checked.
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||07:03
||Choose '''Name and Value''' from the dropdown menu next to it.
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||07:08
||Under the '''Color''' tab, select red.
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||07:12
||Under the '''Style''' tab, select '''dashed line style'''.
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||07:17
||Close the '''Preferences''' box.
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||07:20
||Click on '''Move''' if it is not highlighted.

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||07:24
||Move the labels to see them properly in '''Graphics''' view.

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||07:30
||Now, let us carry out the same steps for segments '''CF1''', '''CF2''', '''DF1''' and '''DF2''' but make them blue.

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||07:39
||Click on '''Move''' if it is not highlighted.

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||07:42
||And move the labels to see them properly in '''Graphics''' view.

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||07:49
||Right-click on points '''A, B, C''' and '''D''' and select '''Trace On''' option.
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||08:03
||Set '''slider k''' at 1.

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||08:07
||Drag '''slider a''' to both ends of the '''slider'''.

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||08:12
||Set first '''k''' at 2.
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||08:17
||Then at 3.
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||08:21
||At 5.
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||08:24
||And finally at 10.
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||08:28
||Observe the traces of hyperbolae for the different values of '''a''' and '''k'''
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||08:34
||Let us look at the equations of hyperbolae.

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||08:38
||Open a new '''GeoGebra''' window.

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||08:41
||In the '''input bar''', type the following line describing the difference of two fractions equal to 1.

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||08:48
||To type the '''caret symbol''', hold the '''Shift''' key down and press 6.

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||08:53
||For the 1st fraction, type the numerator as '''x minus h''' in parentheses '''caret 2'''.

Then type division slash.

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||09:03
||Now, type the denominator of the 1st fraction as '''a caret 2''' followed by '''minus'''.

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||09:11
||For the 2nd fraction, type the numerator as '''y minus k''' in parentheses '''caret 2'''.

Then type division slash.

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||09:01
||Now, type the denominator of the 2nd fraction as '''b caret 2''' followed by '''equals sign 1'''.

Press '''Enter'''.
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||09:31
||A pop-up window asks if you want to create '''sliders''' for '''a, b, h''' and '''k'''.
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||09:38
||Click on '''Create Sliders'''.
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||09:41
||This creates number '''sliders''' for '''h, a, k''' and ''' b'''.
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||09:48
||By default, they go from minus 5 to 5 and are set at 1.

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||09:54
||You can double-click on the '''sliders''' to see their properties.
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||09:58
||A hyperbola appears in '''Graphics''' view.
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||10;02
||Under '''Move Graphics View''', click on '''Zoom Out''' and then in '''Graphics '''view.
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||10:11
||Click on '''Move Graphics View''' and drag '''Graphics''' view to see the hyperbola properly.
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||10:20
||In '''Algebra''' view, note the equation for hyperbola '''c'''.
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||10:25
||Drag the boundary to see it properly.
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||10:29
||Keep track of the equations appearing in '''Algebra''' view as you drag the '''sliders''' from end to end.
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||10:36
||You will see the effects on the shape of hyperbola '''c'''.
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||10:41
||Place the cursor over the equation in '''Algebra''' view.
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||10:46
||Note that '''a''' is associated with the '''x minus h squared''' component of the equation.

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||10:53
||It controls the horizontal movement of hyperbola '''c'''.
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||11:00
||Associated with the '''y minus k squared''' component is '''b'''.

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||11:06
||It controls the vertical movement of hyperbola '''c'''.
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||11:12
||Note that the '''transverse axis''' of hyperbola '''c''' is horizontal like the '''x axis'''.
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||11:19
||Drag '''slider a''' to 2, leaving '''b''' at 1.
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||11:25
||When '''a''' is greater than '''b''', the arms of the hyperbola are closer to the '''x axis'''.
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||11:32
||Note the equation of the hyperbola.
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||11:36
||Drag the boundary to see it properly.
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||11:39
||With '''slider a''' at 2, drag '''slider b''' to 3.
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||11:44
||When '''a''' is less than '''b''', the arms of the hyperbola stretch closer to the '''y axis'''.
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||11:52
||Note the equation of hyperbola '''c'''.
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||11:56
||Drag the boundary to see it properly.
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||12:00
||With '''slider a''' at 2, drag '''slider b''' back to 1.
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||12:06
||Click in and drag '''Graphics''' view to see the hyperbola properly.
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||12:12
||In the '''input bar''', type '''Focus c''' in parentheses and press '''Enter'''.
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||12:20
||Two foci,''' A''' and '''B''', are mapped in '''Graphics''' view.

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||12:25
||Their '''coordinates''' appear in '''Algebra''' view.
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||12:29
||In the '''input bar''', type '''Center c''' in parentheses and press '''Enter'''.
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||12:37
||Center, point '''C''', appears in '''Graphics''' view.

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||12:42
||Its co-ordinates appear in '''Algebra''' view.
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||12:46
||Note that the center has the coordinates '''h comma k'''.
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||12:52
||Drag '''sliders h''' and '''k''' from end to end.
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||12:59
||Note the effects on hyperbola '''c'''.
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||13:02
||In the '''input bar''', type '''Vertex c''' in parentheses and press '''Enter'''.
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||13:11
||Vertices, '''D''' and '''E''', appear on hyperbola '''c'''.

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||13:17
||Let us drag ''' a''' so we can see the vertices clearly.
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||13:23
||Drag the boundary to see '''Graphics''' view properly.
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||13:28
||Click in '''Graphics''' view and drag the background so you can see the hyperbola properly.
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||13:34
||Drag '''slider a''' back to 2.
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||13:38
||Under '''Slider''', click on '''Text''' and click in '''Graphics''' view.
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||13:45
||A text-box opens up.

In the '''Edit''' field, type the following text.
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||13:52
||Press '''Enter''' after each line to go to the next line and

Click '''OK'''.
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||13:58
||Refer to '''additional material''' provided with this '''tutorial''' for these calculations.
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||14:05
||Click on '''Move Graphics View''' and drag the background so you can see the hyperbola.
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||14:13
||Uncheck equation '''c''' and all points and text generated for '''hyperbola c''' in '''Algebra''' view.
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||14:25
||Follow the earlier steps to construct hyperbola '''d''' for these two conditions.
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||14:32
||Let us summarize.
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||14:34
||In this tutorial, we have learnt how to use '''GeoGebra''' to:

Construct a hyperbola

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||14:41
||Look at standard equations and parts of hyperbolae
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||14:45
||As an assignment,

Find all these values.
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||14:53
||Find all these values for these hyperbolae.
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||15:00
||The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.
|-
||15:08
||The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.

For more details, please write to us.
|-
||15:17
||Please post your timed queries on this forum.
|-
||15:21
||'''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
|-
||15:34
||This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off.

Thank you for joining.
|-
|}

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

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