Applications-of-GeoGebra/C2/Conic-Sections-Hyperbola/English-timed
From Script | Spoken-Tutorial
Time | Narration |
00:01 | Welcome to this tutorial on Conic Sections - Hyperbola. |
00:06 | In this tutorial, we will:
Study standard equations and parts of hyperbolae |
00:13 | Learn how to use GeoGebra to construct a hyperbola. |
00:18 | Here I am using:
Ubuntu Linux OS version 14.04 , GeoGebra 5.0.388.0 hyphen d |
00:33 | To follow this tutorial, you should be familiar with
GeoGebra interface |
00:40 | Conic Sections in geometry |
00:43 | For relevant tutorials, please visit our website. |
00:48 | Hyperbola,
Consider two fixed points F1 and F2 called foci. |
00:57 | A hyperbola is the locus of points whose difference of distances from these foci is constant. |
01:07 | In the image, observe that foci of a hyperbola lie along the transverse axis. |
01:15 | They are equidistant from the center which lies on the conjugate axis. |
01:22 | 2b is the length of the conjugate axis. |
01:27 | c is the distance of each focus from the center. |
01:33 | The conjugate axis is perpendicular to the transverse axis. |
01:38 | The hyperbola intersects the transverse axis at the vertices A and B. |
01:46 | a is the distance of each vertex from the center. |
01:52 | The latus recti pass through the foci. |
01:56 | They are perpendicular to the transverse axis. |
02:00 | Be careful to distinguish lengths from letters used for sliders, circles and hyperbolae. |
02:08 | Let us construct a hyperbola in GeoGebra. |
02:13 | I have already opened the GeoGebra interface. |
02:18 | Click on Point tool and click twice in Graphics view. |
02:26 | This creates two points A and B, which will be the foci of our hyperbola. |
02:33 | Right-click on A and choose the Rename option. |
02:39 | In the New Name field, type F1 and click OK. |
02:45 | This will be one of our foci, F1. |
02:49 | Let us rename point B as F2. |
02:53 | Click on Slider tool and click in Graphics view. |
03:00 | A Slider dialog-box appears in Graphics view. |
03:04 | Stay with the default Number selection. |
03:08 | In the Name field, type k. |
03:12 | Set Min value as 0, Max value as 10, increment as 0.1, click OK. |
03:26 | This creates a number slider named k. |
03:30 | Using this slider, k can be changed from 0 to 10. |
03:36 | k will be the difference of the distances of any point on the hyperbola from the foci, F1 and F2. |
03:45 | Drag slider k to 4. |
03:49 | We will create another number slider named a. |
03:54 | Its Min value is 0, Max value is 25,increment is 0.1. |
04:02 | Click on Circle with Center and Radius tool and click on F1. |
04:10 | A text-box appears; type a and click OK. |
04:17 | Drag a to a value between 2 and 3. |
04:23 | A circle c with center F1 and radius a appears. |
04:29 | Drag slider a to 5. |
04:33 | Under Move Graphics View, click on Zoom Out tool. |
04:39 | Click in Graphics view. |
04:42 | Click on Move Graphics View to move the background as required. |
04:48 | Click again on Circle with Center and Radius tool and click on F2. |
04:56 | In the text-box, type a minus k and click OK. |
05:03 | Circle d with center F2 and radius a minus k appears. |
05:10 | Click again on Circle with Center and Radius tool and click on F2. |
05:18 | In the text-box, type a plus k and click OK. |
05:25 | Circle e with center F2 and radius a plus k appears. |
05:32 | Set slider k between 1 and 2, slider a between 3 and 4. |
05:40 | Under Point, click on Intersect. |
05:46 | Then click on circles c and d and circles c and e. |
05:55 | This creates points A, B, C and D. |
06:05 | Under Line, click on Segment and click on points A and F1 to join them. |
06:15 | Then click on points A and F2 to join them. |
06:21 | Similarly, using Segment tool, join B and F1 as well as B and F2. |
06:31 | Click on Move. |
06:34 | Double click on segment AF1 and click on Object Properties. |
06:42 | In the left panel, segment AF1 is already highlighted. |
06:48 | Holding Ctrl Key down, click and highlight segments AF2, BF1 and BF2. |
06:58 | Under the Basic tab, make sure Show Label is checked. |
07:03 | Choose Name and Value from the dropdown menu next to it. |
07:08 | Under the Color tab, select red. |
07:12 | Under the Style tab, select dashed line style. |
07:17 | Close the Preferences box. |
07:20 | Click on Move if it is not highlighted. |
07:24 | Move the labels to see them properly in Graphics view. |
07:30 | Now, let us carry out the same steps for segments CF1, CF2, DF1 and DF2 but make them blue. |
07:39 | Click on Move if it is not highlighted. |
07:42 | And move the labels to see them properly in Graphics view. |
07:49 | Right-click on points A, B, C and D and select Trace On option. |
08:03 | Set slider k at 1. |
08:07 | Drag slider a to both ends of the slider. |
08:12 | Set first k at 2. |
08:17 | Then at 3. |
08:21 | At 5. |
08:24 | And finally at 10. |
08:28 | Observe the traces of hyperbolae for the different values of a and k |
08:34 | Let us look at the equations of hyperbolae. |
08:38 | Open a new GeoGebra window. |
08:41 | In the input bar, type the following line describing the difference of two fractions equal to 1. |
08:48 | To type the caret symbol, hold the Shift key down and press 6. |
08:53 | For the 1^{st} fraction, type the numerator as x minus h in parentheses caret 2.
Then type division slash. |
09:03 | Now, type the denominator of the 1^{st} fraction as a caret 2 followed by minus. |
09:11 | For the 2^{nd} fraction, type the numerator as y minus k in parentheses caret 2.
Then type division slash. |
09:01 | Now, type the denominator of the 2^{nd} fraction as b caret 2 followed by equals sign 1.
Press Enter. |
09:31 | A pop-up window asks if you want to create sliders for a, b, h and k. |
09:38 | Click on Create Sliders. |
09:41 | This creates number sliders for h, a, k and b. |
09:48 | By default, they go from minus 5 to 5 and are set at 1. |
09:54 | You can double-click on the sliders to see their properties. |
09:58 | A hyperbola appears in Graphics view. |
10;02 | Under Move Graphics View, click on Zoom Out and then in Graphics view. |
10:11 | Click on Move Graphics View and drag Graphics view to see the hyperbola properly. |
10:20 | In Algebra view, note the equation for hyperbola c. |
10:25 | Drag the boundary to see it properly. |
10:29 | Keep track of the equations appearing in Algebra view as you drag the sliders from end to end. |
10:36 | You will see the effects on the shape of hyperbola c. |
10:41 | Place the cursor over the equation in Algebra view. |
10:46 | Note that a is associated with the x minus h squared component of the equation. |
10:53 | It controls the horizontal movement of hyperbola c. |
11:00 | Associated with the y minus k squared component is b. |
11:06 | It controls the vertical movement of hyperbola c. |
11:12 | Note that the transverse axis of hyperbola c is horizontal like the x axis. |
11:19 | Drag slider a to 2, leaving b at 1. |
11:25 | When a is greater than b, the arms of the hyperbola are closer to the x axis. |
11:32 | Note the equation of the hyperbola. |
11:36 | Drag the boundary to see it properly. |
11:39 | With slider a at 2, drag slider b to 3. |
11:44 | When a is less than b, the arms of the hyperbola stretch closer to the y axis. |
11:52 | Note the equation of hyperbola c. |
11:56 | Drag the boundary to see it properly. |
12:00 | With slider a at 2, drag slider b back to 1. |
12:06 | Click in and drag Graphics view to see the hyperbola properly. |
12:12 | In the input bar, type Focus c in parentheses and press Enter. |
12:20 | Two foci, A and B, are mapped in Graphics view. |
12:25 | Their coordinates appear in Algebra view. |
12:29 | In the input bar, type Center c in parentheses and press Enter. |
12:37 | Center, point C, appears in Graphics view. |
12:42 | Its co-ordinates appear in Algebra view. |
12:46 | Note that the center has the coordinates h comma k. |
12:52 | Drag sliders h and k from end to end. |
12:59 | Note the effects on hyperbola c. |
13:02 | In the input bar, type Vertex c in parentheses and press Enter. |
13:11 | Vertices, D and E, appear on hyperbola c. |
13:17 | Let us drag a so we can see the vertices clearly. |
13:23 | Drag the boundary to see Graphics view properly. |
13:28 | Click in Graphics view and drag the background so you can see the hyperbola properly. |
13:34 | Drag slider a back to 2. |
13:38 | Under Slider, click on Text and click in Graphics view. |
13:45 | A text-box opens up.
In the Edit field, type the following text. |
13:52 | Press Enter after each line to go to the next line and
Click OK. |
13:58 | Refer to additional material provided with this tutorial for these calculations. |
14:05 | Click on Move Graphics View and drag the background so you can see the hyperbola. |
14:13 | Uncheck equation c and all points and text generated for hyperbola c in Algebra view. |
14:25 | Follow the earlier steps to construct hyperbola d for these two conditions. |
14:32 | Let us summarize. |
14:34 | In this tutorial, we have learnt how to use GeoGebra to:
Construct a hyperbola |
14:41 | Look at standard equations and parts of hyperbolae |
14:45 | As an assignment,
Find all these values. |
14:53 | Find all these values for these hyperbolae. |
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15:34 | This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. |