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

Madhurig at 06:48, 11 September 2019

2019-09-11T06:48:26Z

Snehalathak at 07:54, 30 August 2018

2018-08-30T07:54:21Z

Snehalathak at 07:20, 30 August 2018

2018-08-30T07:20:36Z

Madhurig at 06:17, 30 August 2018

2018-08-30T06:17:45Z

Madhurig at 07:30, 14 August 2018

2018-08-14T07:30:33Z

Vidhya at 06:59, 13 August 2018

2018-08-13T06:59:25Z

Madhurig at 10:14, 25 July 2018

2018-07-25T10:14:00Z

Madhurig at 08:04, 25 July 2018

2018-07-25T08:04:41Z

Madhurig at 07:54, 25 July 2018

2018-07-25T07:54:50Z

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Vidhya: Created page with "{|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Conic Sections - Ellipse''' |- | | '''Sl..."

2018-07-23T06:31:03Z

Created page with "{|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Conic Sections - Ellipse''' |- | | '''Sl..."

New page

{|border=1
| | '''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'''

[[Image:]]

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.
| | 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 center '''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 and 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''' and choose '''Rename''' option.
| | Right-click on '''A''' and choose the '''Rename''' option.
|-
| | Type '''F1''' in the '''New Name''' field and 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 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.

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'''.
|-
| |
| | 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 center '''F1''' and radius '''a'''.
| | A circle '''c''' with center '''F1''' and radius '''a''' appears.
|-
| | 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 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''' and radius '''k-a''' 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.
| | Drag '''slider k''' to 5 and '''slider a''' to 2.
|-
| | 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''' and 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''' and '''A''' to join them.
| | Click on points '''F1''' and '''A''' to join them.
|-
| | 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'''.
|-
| | 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 and 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''' 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'''.
| | Note the locus of points traced by '''A''' and '''B'''.

These traced points are all equidistant from '''F1''' and '''F2''' (the foci).

They lie on ellipses for which '''F1''' and '''F2''' are foci.
|-
| | 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''' 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 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 and move '''F1''' and '''F2''' to different positions in '''Graphics View'''.
| | Click on '''Move''' tool and move '''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.

Center '''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 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'''.
| | 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.
|-
| | Drag boundary to see the equation properly.
| | Drag the boundary to see it properly.
|-
| | 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''' and 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.
|-
| | Click on '''Text''' tool under '''Slider''' tool and 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
| | '''Text box for ellipse c'''

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.
|-
| |
| | 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 8'''

'''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 9'''

'''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 10'''

'''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 11'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.
|-
| | '''Slide Number 12'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates.

For more details, please write to us.
|-
| | '''Slide Number 13'''

'''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 14'''

'''Acknowledgement'''
| | '''Spoken Tutorial''' Project is funded by NMEICT, MHRD, Government of India.

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

Thank you for joining.
|-
|}

@@ Line 410: / Line 410: @@
 || Point to the equation in '''Algebra''' view.
 || Note also the change in the equation in '''Algebra''' view.
 |-
 ||Point to the ellipse.

@@ Line 241: / Line 241: @@
 |-
 || Point to the traces of '''A''' and '''B''' and to '''A''' and '''B'''.
-|| Note the locus of points traced by points'''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.
+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.
+They lie on ellipses for which points ''F1''' and '''F2''' are foci.
 |-
 || Right-click on '''sliders a''' >> '''k''' >> uncheck '''Animation On''' option.
@@ Line 267: / Line 267: @@
 || Click in and move '''Graphics''' view slightly to erase the trace points.
 |-
-|| Click on the '''Move''' tool >> 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'''.
@@ Line 368: / Line 368: @@
 |-
 || Type '''Vertex(c)''' in the '''input bar''' >> press '''Enter'''.
-|| In the '''input bar''', type '''Vertex c'''  in parentheses >> 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.
@@ Line 393: / Line 393: @@
 '''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.

@@ Line 241: / Line 241: @@
 |-
 || 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''' >> '''k''' >> uncheck '''Animation On''' option.

@@ Line 133: / Line 133: @@
 |-
 || 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 centre '''F1''' >> radius '''a'''.
@@ Line 151: / Line 151: @@
 |-
 || 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''' and in '''Graphics''' view.
 |-
 || Click on '''Move Graphics View''' >> drag '''Graphics''' view.

@@ Line 37: / Line 37: @@
 Conic sections in geometry
 For relevant tutorials, please visit our website.
@@ Line 45: / Line 44: @@
 '''Ellipse'''
-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.
@@ Line 282: / Line 281: @@
 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'''.
@@ Line 288: / Line 287: @@
 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'''.
@@ Line 312: / Line 311: @@
 || 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.
@@ Line 357: / Line 356: @@
 |-
 || Type '''Focus(c)''' in the '''input bar''' and press '''Enter'''.
-|| In the '''input bar''', type '''Focus c in parentheses''' 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.
@@ Line 363: / Line 362: @@
 |-
 || Type '''Center(c)''' in the '''input bar''' >> press '''Enter'''.
-|| In the '''input bar''', type '''Center c in parentheses''' and 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.
@@ Line 369: / Line 368: @@
 |-
 || Type '''Vertex(c)''' in the '''input bar''' >> press '''Enter'''.
-|| In the '''input bar''', type '''Vertex c in parentheses''' >> press '''Enter'''.
+|| In the '''input bar''', type '''Vertex c'''  in parentheses >> press '''Enter'''.
 Vertices '''D''' and '''E''' appear at the ends of the major axis.
@@ Line 414: / Line 413: @@
 || Note also the change in the equation in '''Algebra''' view.
 |-
-||Point to the elipse.
+||Point to the ellipse.
 || Calculate eccentricity and length of latus recti, major and minor axes for this ellipse.
 |-

@@ Line 355: / Line 355: @@
 Note the equation of the ellipse.
 |-
 || Type '''Focus(c)''' in the '''input bar''' and press '''Enter'''.

@@ Line 21: / Line 21: @@
 '''System Requirement'''
-|| Here I am using:
+|| Here, I am using:
 '''Ubuntu Linux ''' OS version 14.04
@@ Line 56: / Line 56: @@
 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'''
 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'''.
@@ Line 71: / Line 73: @@
 |-
 ||
-|| 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.
@@ Line 120: / Line 122: @@
 '''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'''.
 |-
@@ Line 134: / Line 136: @@
 || A '''text-box''' appears; in the radius '''Name''' field, type '''a''' and click '''OK'''.
 |-
-|| Point to the circle with center '''F1''' >> 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 >> '''slider k''' to 5.
@@ Line 148: / Line 150: @@
 || 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.