Applications-of-GeoGebra/C2/Conic-Sections-Ellipse/English-timed
From Script | Spoken-Tutorial
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 |
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. |
00:55 | These fixed points are called the foci. |
01:00 | Observe the centre O, foci F1 and F2. |
01:08 | Vertices A and B are at the ends of the major axis AB. |
01:15 | Co-vertices C and D are at the ends of the minor axis CD. |
01:23 | Two latus recti pass through the foci. |
01:27 | Axes lengths 2a and 2b and distance between the foci 2c are shown in the figure. |
01:37 | Be careful to distinguish length, from letters used for sliders, circles and ellipses. |
01:45 | Let us construct an ellipse in GeoGebra. |
01:49 | I have already opened the GeoGebra interface. |
01:54 | Click on Point tool and click twice in Graphics view. |
02:03 | This creates two points A and B, which will be the foci of our ellipse. |
02:10 | Right-click on A and choose the Rename option. |
02:16 | In the New Name field, type F1 and click OK. |
02:23 | This will be one of our foci, F1. |
02:28 | Let us rename B as F2. |
02:32 | Click on Slider tool and click in Graphics view. |
02:40 | Slider dialog box appears in Graphics view. |
02:45 | Stay with the default Number selection and in the Name field, type k. |
02:53 | Set Min value as 0, Max value as 10, increment as 0.1.
Click OK. |
03:07 | This creates a number slider named k. |
03:12 | Slider k can be changed from 0 to 10. |
03:17 | k will be the sum of the distances of any point on the ellipse from the foci F1 and F2. |
03:26 | We will create another number slider named a. |
03:31 | Its Min value is 0, Max value is 10, increment is 0.1." |
03:39 | Click on Circle with Center and Radius tool and click on F1. |
03:49 | A textbox appears. In the Radius name field, type a and click OK. |
04:00 | A circle c with centre F1 and radius a appears. |
04:08 | Drag slider a to 2 and slider k to 5. |
04:16 | Click again on Circle with Center and Radius tool and click on F2. |
04:26 | In the text box that appears, type k minus a and click OK. |
04:35 | A circle d with center F2 and radius k minus a appears in Graphics view. |
04:44 | Under Move Graphics View, click on Zoom Out and in Graphics view. |
04:53 | Click on Move Graphics View and drag Graphics view. |
05:01 | Under Point, click on Intersect tool. |
05:06 | Click on the two circles c and d. |
05:12 | This creates points A and B. |
05:16 | Under Line, click on Segment tool. |
05:21 | Click on points F1 and A to join them. |
05:27 | Next, click on points A and F2 to join them. |
05:33 | Click on Move. |
05:36 | Double click on Segment AF1. |
05:40 | Click on Object Properties to open the Preferences dialog box. |
05:45 | Segment AF1 is already highlighted in the left panel. |
05:51 | Holding Ctrl key down, click and highlight Segment AF2 as well. |
05:58 | Under the Basic tab, make sure that Show Label is selected. |
06:04 | Pull down the drop down menu next to the Show Label check box. |
06:09 | Select Name and Value. |
06:12 | Under the Color tab, select red. |
06:16 | Under the Style tab, choose dashed line style. |
06:23 | Close the Preferences dialog box. |
06:26 | Draw Segments BF1 and BF2.
Make them dashed and blue. |
06:31 | Make sure that the Move tool is highlighted. |
06:39 | Move the labels so you can see them properly. |
06:45 | Note that the sum of the segment lengths from both foci to each intersection point is equal to k. |
06:55 | Right-click on A and B and check Trace On option. |
07:04 | In Algebra view, uncheck circles c and d to hide the circles. |
07:10 | Right-click on slider a and check Animation On option. |
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. |
07:39 | They lie on ellipses for which points F1' and F2 are foci. |
07:45 | Right-click on sliders a and k and uncheck Animation On option. |
07:56 | Drag sliders a and k to different values to see more traces of ellipses. |
08:06 | Set slider k between 9 and 10 and slider a between 5 and 6. |
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. |
08:27 | Note the same fact for Segments BF1 and BF2. |
08:33 | Click in and move Graphics view slightly to erase the trace points. |
08:39 | Click on Move tool and move points F1 and F2 to different positions in Graphics View. |
08:50 | Values can be changed on sliders a and k to see various ellipses. |
08:58 | Let us look at the equations of ellipses in a new GeoGebra window. |
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. |
09:18 | For the 1^{st} fraction, type the numerator as x minus h in parentheses caret 2.
Then type division slash. |
09:30 | Now, type the denominator of the 1^{st} fraction as a caret 2 followed by plus. |
09:37 | For the 2^{nd} 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 2^{nd} 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. |
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. |
10:39 | In Algebra view, note the equation for circle c. |
10:44 | Drag the boundary to see it properly. |
10:48 | Under Move Graphics View, click on Zoom Out tool and in Graphics view. |
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. |
11:09 | Place the cursor on equation c in Algebra view. |
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. |
11:28 | Associated with the y minus k squared component is b. |
11:34 | Observe how b controls the vertical axis of the ellipse. |
11:41 | Drag slider a to 2 and b to 1. |
11:48 | When a is greater than b, the major axis of the ellipse is horizontal.
Note the equation of the ellipse. |
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. |
12:16 | In the input bar, type Center c in parentheses and press Enter. |
12:25 | Center C appears in Graphics view and its co-ordinates appear in Algebra view. |
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. |
12:48 | Co-vertices F and G appear at the ends of the minor axis. |
12:54 | Under Slider, click on Text tool and click in Graphics view. |
13:02 | A text-box opens up. |
13:05 | In the Edit field, type the following text. |
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. |
13:29 | Note the effects on the shape of ellipse c and the change in directions of major and minor axes. |
13:38 | Note also the change in the equation in Algebra view. |
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. |
14:37 | Find all these values for these ellipses. |
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15:17 | This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. |