|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.|
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.
|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.
|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.
|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.|
|14:42|| The video at the following link summarizes the Spoken Tutorial project.
Please download and watch it.
|14:51|| The Spoken Tutorial Project team conducts workshops and gives certificates.
For more details, please write to us.
|15:00||Please post your timed queries on this forum.|
|15:04|| Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link.
|15:17|| This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining.