Difference between revisions of "Applications-of-GeoGebra/C2/Conic-Sections-Hyperbola/English"
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||Set '''slider k''' at 2 and drag '''slider a''' to both ends. | ||Set '''slider k''' at 2 and drag '''slider a''' to both ends. | ||
| − | || | + | ||Set first '''k''' at 2. |
|- | |- | ||
||Set '''slider k''' at 3 and drag '''slider a''' to both ends. | ||Set '''slider k''' at 3 and drag '''slider a''' to both ends. | ||
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||Point to the equations appearing in '''Algebra''' view. | ||Point to the equations appearing in '''Algebra''' view. | ||
| − | ||Keep track of the equations appearing in '''Algebra''' view as you | + | ||Keep track of the equations appearing in '''Algebra''' view as you drag the '''sliders''' from end to end. |
|- | |- | ||
||Point to hyperbola '''c'''. | ||Point to hyperbola '''c'''. | ||
Latest revision as of 16:28, 31 August 2018
| Visual Cue | Narration |
| Slide Number 1
Title Slide |
Welcome to this tutorial on Conic Sections - Hyperbola. |
| Slide Number 2
Learning Objectives |
In this tutorial, we will:
Study standard equations and parts of hyperbolae Learn how to use GeoGebra to construct a hyperbola. |
| 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 & 6
Hyperbola [[Image:]] Consider two fixed points F1and F2 called foci. A hyperbola is the locus of points whose difference of distances from these foci is constant. |
Hyperbola,
Consider two fixed points F1 and F2 called foci. A hyperbola is the locus of points whose difference of distances from these foci is constant. In the image, observe that foci of a hyperbola lie along the transverse axis. They are equidistant from the center which lies on the conjugate axis. 2b is the length of the conjugate axis. c is the distance of each focus from the center. The conjugate axis is perpendicular to the transverse axis. The hyperbola intersects the transverse axis at the vertices A and B. a is the distance of each vertex from the center. The latus recti pass through the foci. They are perpendicular to the transverse axis. |
| Be careful to distinguish lengths from letters used for sliders, circles and hyperbolae. | |
| Show the GeoGebra window. | Let us construct a hyperbola in GeoGebra.
I have already opened the GeoGebra interface. |
| Click on Point tool >> click twice in Graphics view.
Point to A and B. |
Click on Point tool and click twice in Graphics view.
This creates two points A and B, which will be the foci of our hyperbola. |
| 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 point 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. | A Slider dialog-box appears in Graphics view. |
| Stay with the default Number radio-button selection.
In the Name field, type k. |
Stay with the default Number selection.
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 the slider k from 1 to 10. | Using this slider, k can be changed from 0 to 10. |
| Point to slider k. | k will be the difference of the distances of any point on the hyperbola from the foci, F1 and F2. |
| Drag slider k to 4. | Drag slider k to 4. |
| We will create another number slider named a. | |
| Point to slider a. | Its Min value is 0, Max value is 25,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 text box; type a >> click OK. | A text-box appears; type a and click OK. |
| Drag a to a value between 2 and 3. | Drag a to a value between 2 and 3. |
| Point to the circle c with center F1 and radius a. | A circle c with center F1 and radius a appears. |
| Drag slider a to 5. | Drag slider a to 5. |
| Under Move Graphics View, click on Zoom Out tool. | Under Move Graphics View, click on Zoom Out tool. |
| Click in Graphics view. | Click in Graphics view. |
| Click on Move Graphics View tool to move the background. | Click on Move Graphics View to move the background as required. |
| 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 a-k >> click OK. | In the text-box, type a minus k and click OK. |
| Point to the circle d with center F2 and radius a-k. | Circle d with center F2 and radius a minus k appears. |
| 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 a+k >> click OK. | In the text-box, type a plus k and click OK. |
| Point to the circle e with center F2 and radius a+k. | Circle e with center F2 and radius a plus k appears. |
| Set slider k between 1 and 2, slider a between 3 and 4. | Set slider k between 1 and 2, slider a between 3 and 4. |
| Click on Intersect tool under Point tool.
Click on circles cand dand circles cand e. |
Under Point, click on Intersect.
Then click on circles c and d and circles c and e. |
| Point to points A, B, C and D. | This creates points A, B, C and D. |
| Under Line tool, click on Segment tool >> click on points A and F1
Click on points A and F2. |
Under Line, click on Segment and click on points A and F1 to join them.
Then click on points A and F2 to join them. |
| Click on Segment tool >> join the points B and F1 >> join B and F2. | Similarly, using Segment tool, join B and F1 as well as B and F2. |
| Click on Move tool. | Click on Move. |
| Double click on segment AF1 >> Object Properties. | Double click on segment AF1 and click on Object Properties. |
| Point to the left panel and to highlighted segment AF1. | In the left panel, segment AF1 is already highlighted. |
| Holding Ctrl Key down, click and highlight segments AF2, BF1 and BF2. | Holding Ctrl Key down, click and highlight segments AF2, BF1 and BF2. |
| Under Basic tab, make sure Show Label is checked. | Under the Basic tab, make sure Show Label is checked. |
| Choose Name and Value from the dropdown menu next to it. | Choose Name and Value from the dropdown menu next to it. |
| Under Color tab, select red. | Under the Color tab, select red. |
| Under Style tab, select dashed line style. | Under the Style tab, select dashed line style. |
| Close Preferences box. | Close the Preferences box. |
| Click on Move tool >> move the labels properly in Graphics view. | Click on Move if it is not highlighted.
Move the labels to see them properly in Graphics view. |
| Now, let us carry out the same steps for segments CF1, CF2, DF1 and DF2 but make them blue. | |
| Click on Move tool >> move the labels properly in Graphics view. | Click on Move if it is not highlighted.
And move the labels to see them properly in Graphics view. |
| Right-click on points A, B, C and D >> select Trace On option. | Right-click on points A, B, C and D and select Trace On option. |
| Set slider k at 1.
Drag slider a to both ends of slider. |
Set slider k at 1.
Drag slider a to both ends of the slider. |
| Set slider k at 2 and drag slider a to both ends. | Set first k at 2. |
| Set slider k at 3 and drag slider a to both ends. | Then at 3. |
| Set slider k at 5 and drag slider a again to both ends. | At 5. |
| Set slider k at 10 and drag slider a yet again to both ends. | And finally at 10. |
| Point to the traces of hyperbolae for different values of a and k. | Observe the traces of hyperbolae for the different values of a and k. |
| Let us look at the equations of hyperbolae. | |
| In the input bar, type (x-h)^2/a^2-(y-k)^2/b^2=1 and press Enter.
To type the caret symbol, hold the Shift key down and press 6. |
Open a new GeoGebra window.
In the input bar, type the following line describing the difference of two fractions equal to 1.
Then type division slash. Now, type the denominator of the 1st fraction as a caret 2 followed by minus. 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 popup 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. |
| Double-click on the sliders to see their properties. | 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 the hyperbola in Graphics view. | A hyperbola appears in Graphics view. |
| Under Move Graphics View, click on Zoom Out tool >> click in Graphics view. | Under Move Graphics View, click on Zoom Out and then in Graphics view. |
| Click on Move Graphics View >> drag Graphics view to see hyperbola properly. | Click on Move Graphics View and drag Graphics view to see the hyperbola properly. |
| Point to the equation for hyperbola c in Algebra view. | In Algebra view, note the equation for hyperbola c. |
| Drag boundary to left of Slider tool see equation properly. | Drag the boundary to see it properly. |
| Point to the equations appearing in Algebra view. | Keep track of the equations appearing in Algebra view as you drag the sliders from end to end. |
| Point to hyperbola c. | You will see the effects on the shape of hyperbola c. |
| Place the cursor over the equation in Algebra view. | Place the cursor over the equation in Algebra view. |
| Point to slider a and hyperbola c.
Drag slider a. |
Note that a is associated with the x minus h squared component of the equation.
It controls the horizontal movement of hyperbola c. |
| Point to slider b and hyperbola c.
Drag slider b. |
Associated with the y minus k squared component is b.
It controls the vertical movement of hyperbola c. |
| Point to hyperbola c. | Note that the transverse axis of hyperbola c is horizontal like the x axis. |
| Drag slider a to 2, leaving b at 1. | Drag slider a to 2, leaving b at 1. |
| Point to hyperbola c. | When a is greater than b, the arms of the hyperbola are closer to the x axis. |
| Point to equation of hyperbola c. | Note the equation of the hyperbola. |
| Drag boundary to see it properly. | Drag the boundary to see it properly. |
| Point to slider at 2.
Drag slider b to 3. |
With slider a at 2, drag slider b to 3. |
| Point to sliders a and b, and hyperbola c. | When a is less than b, the arms of the hyperbola stretch closer to the y axis. |
| Point to equation of hyperbola c. | Note the equation of hyperbola c. |
| Drag boundary to see it properly. | Drag the boundary to see it properly. |
| With slider a at 2, drag slider b to 1. | With slider a at 2, drag slider b back to 1. |
| Click in and drag Graphics view to see hyperbola properly. | Click in and drag Graphics view to see the hyperbola properly. |
| Type Focus(c) in the input bar >> press Enter. | In the input bar, type Focus c in parentheses and press Enter. |
| Point to A and B in Graphics view.
Point to the co-ordinates in Algebra view. |
Two foci, A and B, are mapped in Graphics view.
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
Point the co-ordinates in Algebra view. |
Center, point C, appears in Graphics view.
Its co-ordinates appear in Algebra view. |
| Point to the co-ordinates (h, k) of center C in Algebra view. | Note that the center has the coordinates h comma k. |
| Drag sliders h and k from end to end. | Drag sliders h and k from end to end. |
| Point to hyperbola c. | Note the effects on hyperbola c. |
| Type Vertex(c) in the input bar >> press Enter. | In the input bar, type Vertex c in parentheses and press Enter. |
| Point to D and E.
Drag slider a to ~0.5 so we can see the vertices clearly. |
Vertices, D and E, appear on hyperbola c.
Let us drag slider a so we can see the vertices clearly. |
| Drag boundary to see Graphics view properly. | Drag the boundary to see Graphics view properly. |
| Click in and drag Graphics view to see hyperbola. | Click in Graphics view and drag the background so you can see the hyperbola properly. |
| Drag slider a back to 2. | Drag slider a back to 2. |
| Click on Text tool under Slider tool >> click in Graphics view. | Under Slider, click on Text and click in Graphics view. |
| Point to the text box.
In Edit field, type the following text. |
A text-box opens up.
In the Edit field, type the following text. |
| Slide Number 7.
Text box for Hyperbola c Transverse axis 2a = 4 c = 2.24 Conjugate axis 2b = 2.018 e = 1.12 latus rectum = 1.018 |
Press Enter after each line to go to the next line and
Click OK. |
| Refer to additional material provided with this tutorial for these calculations. | |
| Click on Move Graphics View and drag the background to see the hyperbola. | Click on Move Graphics View and drag the background so you can see the hyperbola. |
| Uncheck equation c and all points and text generated for hyperbola c in Algebra view. | Uncheck equation c and all points and text generated for hyperbola c in Algebra view. |
| Show screenshots of hyperbola d for a=2, b=1 and a=2, b=3. | Follow the earlier steps to construct hyperbola d for these two conditions. |
| Let us summarize. | |
| Slide Number 8
Summary |
In this tutorial, we have learnt how to use GeoGebra to:
Construct a hyperbola Look at standard equations and parts of hyperbolae |
| Slide Number 9
Assignment Construct hyperbolae with: Foci (± 3, 0) and vertices (± 2, 0) Foci (0, ± 5) and vertices (0, ± 3) Find their centres and equations. Calculate eccentricity and length of latus recti, transverse and conjugate axes. |
As an assignment,
Find all these values. |
| Slide Number 10
Assignment Find the coordinates of the foci, vertices and eccentricity for these hyperbolae. Also calculate length of the latus rectum and transverse and conjugate axes. x2/16 - y2/9 = 1 49y2 – 16x2 = 784 |
Find all these values for these hyperbolae. |
| 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. |
