Difference between revisions of "Applications-of-GeoGebra/C2/Conic-Sections-Ellipse/English"
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|| Point to the equation in '''Algebra''' view. | || Point to the equation in '''Algebra''' view. | ||
|| Note also the change in 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. | ||Point to the ellipse. |
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 1^{st} fraction, type the numerator as x minus h in parentheses caret 2. Then type division slash. Now, type the denominator of the 1^{st} fraction as a caret 2 followed by plus. For the 2^{nd} fraction, type the numerator as y minus k, in parentheses caret 2. Then type division slash. Now, type the denominator of the 2^{nd} 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: x^{2}/4 + y^{2}/25 = 1 36x^{2} + 4y^{2} = 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 |
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. |