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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 1st fraction, type the numerator as x minus h in parentheses caret 2.

Then type division slash.

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.
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.

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