Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.


 Maud Harper
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1 Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd implicit equtions of n ellipse will be generted, s will two importnt properties of the ellipse: tht the sum of the distnces from ny point to the foci is equl to the mjor dimeter, nd the Pythgoren Property of Ellipses. Mth Objectives Understnd the geometric definition of n ellipse. Generte the prmetric nd implicit equtions for n ellipse. Locte the foci, given the eqution of n ellipse. Discover reltionships between the prmeters of n ellipse. Technology Objectives Use Geometry Expressions to crete more complex locus of points. Find evidence for equivlence using Geometry Expressions. Mth Prerequisites Pythgoren Theorem Trnsltions Prmetric functions nd implicit equtions Sine nd Cosine Technology Prerequisites Knowledge of Geometry Expressions from previous lessons. Mterils Computers, with Geometry Expressions Sltire Softwre Incorported Pge 1 of 10
2 Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes Overview for the Techer 1. Digrm 1 represents typicl result for question 1. If students re getting semicircles with center t point F1, they re creting locus in terms of t insted of d. Only hlf n ellipse is shown becuse the entire ellipse cnnot be generted s function of d. Points below the foci will hve the sme distnces s points bove. An ellipse is like circle in tht it is curve bsed on distnces from fixed points. It is different in tht it hs different rdii. Digrm 1 d F1 Digrm 2 P d+t F2 2. An pproprite result would be X = cosθ X= cos(t) Y = bsinθ. Trnsposing the sine nd Y=b sin(t) cosine functions will hve no effect on the finl curve. Note tht the trnsposed version is lso the smple prmetric function given by the softwre. Using the trnsposed version yields the sme results, grphiclly. It just chnges the strting plce nd direction tht the ellipse is grphed. Encourge students to chnge vlues for nd b to get n ellipse tht is not just circle. A smple is shown in Digrm 2 3. Some ssumptions re mde here bout the symmetricl nture of n ellipse. You my wish to explore these ssumptions with the clss t this time. Desired solutions:. The distnce from F1 to P is 2 m or 2c + m b. The distnce from F2 to P is m c. The sum is therefore 2 or 2c + 2m. d. The width of the ellipse is 2 e. t = 2. Remind students tht they need to type 2*. 4. The constrint from F2 to P is 2 d. Digrm 3 shows expected results. Digrm 3 d F1 O 2 P 2 d F 4 X= cos(t) Y=b sin(t) 2008 Sltire Softwre Incorported Pge 2 of 10
3 5. Desired solutions:. The sum of the distnces is 2. b. The distnce from F2 to P is equl to, since the three points form n isosceles tringle. c. Using the Pythgoren theorem, 2 = b 2 + c 2, so c = b d. Point P will now pper to be on the ellipse., s shown in Digrm In both instnces, the implicit eqution will be Y + X b b = 0 Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes Digrm 4 F1 d 2 b 2 O 2 P 2 b 2 2 d F2 2 4 X= cos(t) Y=b sin(t) 7. The generl prmetric function tht is X = u0 + cos( T ) generted is Y = v0 + bsin( T ) Digrm 5 u 0 A v B The implicit formul is Y + X b b 2Xb u + b u 2Y v + v = 0 8. Steps re s follows: 4 X= cos(t) Y=b sin(t) X=u 0 + cos(t) z 0 Y=v 0 +b sin(t) z 1 Y 2 2 +X 2 b 22 b 2 X b 2 u 0 +b 2 u 2 0 Y 2 v v 2 0 =0 Y + X b b 2Xb u + b u 2Y v + v = ( ) ( ) ( ) ( ) X b Xb u + b u + Y Y v + v = b b X Xu + u + Y Yv + v = b ( ) ( ) = b X u Y v b ( ) ( ) = b X u Y v b b b b ( X u ) ( Y v ) = 1 b 9. Results s follows:. If = b, then the result is circle. b. If < b, then the ellipse is tller thn it is wide. The foci lie on verticl line rther thn 2 horizontl line. The Pythgoren Property would then be b = + c 2008 Sltire Softwre Incorported Pge 3 of 10
4 Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes 10. Summry: X = u0 + cos( T ) The generl prmetric form of the eqution of n ellipse is Y = v0 + bsin( T ) where (u0, v0) is the center of the ellipse nd nd b re the rdii of the ellipse. + = 1 ( X u ) ( ) 0 Y v0 The generl implicit form ot the eqution of n ellipse is where (u0, v0) is the center of the ellipse. If > b,then 2 is the mjor dimeter nd 2b is the minor dimeter. If b <, then 2b is the mjor dimeter nd 2 is the minor dimeter. The sum of the distnces from ny point on the ellipse to the two foci is 2 b The distnce from the center of the ellipse to either focus follows the eqution 2 c = + b 2008 Sltire Softwre Incorported Pge 4 of 10
5 Nme: Dte: The Ellipse In the lst lesson, you found equtions for circle: the locus of points equidistnt from fixed point. An ellipse is defined s ll the points such tht the sum of the distnce from two fixed points is constnt.. Ech of the two points is clled focus (the plurl of focus is foci ). 1. Crete n ellipse using the definition bove. Open new Geometry Expressions drwing. Crete two points, nd nme them F1 nd F2. Constrin the coordintes of these points. Digrm 1 P d t d F1 D + (t d) = constnt F2 Crete third point, nd nme it P. Constrin the distnce from P to F1 to be d. Constrin the distnce from P to F2 to be t d (see Digrm 1 to understnd why!) Lock the vlue of t in the Vrible Pnel, but keep d unlocked. Find the locus of P with prmeter d. If you drg P round, you cn see tht the locus forms prt of n ellipse. To get more of the ellipse: Double click on the curve. Chnge the Strt Vlue nd End Vlue for d The most you cn get is hlf of the ellipse, becuse the softwre ssumes you know which side P is on you drew it there! To get the rest of the ellipse: Drw line segment from F1 to F2. Select the locus curve. Click on construct reflection nd then click on the segment. Some of the ellipse my still be missing. How is n ellipse like circle? How is it different? Sketch of the ellipse 2008 Sltire Softwre Incorported
6 Before continuing, mke sure Geometry Express is set to Rdins. In the Edit Menu Select preferences. Click on the Mth icon t the left. Under Mth, chnge Angle Mode to rdins. 2. Recll tht the generl prmetric eqution for circle with the center t the origin is X = rcos( T ). Predict the generl prmetric eqution for n ellipse. Y = rsin( T ) Test your prediction with Geometry Expressions Open new Geometry Expressions drwing. Click on the Function tool in the Drw tool pnel. Type in your prediction to see if you re right. Mke dditionl guesses if you need to. 3. The curve you creted in prt 2 looks like n ellipse, but is it relly n ellipse? If it is, we will be ble to find its foci, nd the constnt sum of distnces from the foci. Digrm 2. In Digrm 2, how fr is it from focus F1 to point P? b. How fr is it from focus F2 to point P? c. Wht is the sum of distnces from P to the two foci? m F1 F2 P c c m d. Wht is the horizontl width of the ellipse? e. Write n expression for your nswer to prt c, in terms of distnce. 4. Open new Geometry Expressions drwing, nd crete this prmetric function: X = *cos( T) Y = b*sin( T) Use the Vrible Tool Pnel to chnge the vlues of nd b so tht is greter thn b. Lock vrible nd b Sltire Softwre Incorported
7 Add three points to your Geometry Expressions drwing. First, turn on the xes. Drw F1 nd F2 on the xxis. Drw P so tht it is not on either xis, nor is it on the curve. Constrin the distnce from F1 to P to be d. Wht should the constrint from F2 to P be? Review the results from prt 3 to help you decide. 5. Refer to Digrm 3 to find the positions of F1 nd F2. The tringle shown is n isosceles tringle, with P t the vertex. Digrm 3 P. If P is on the ellipse, wht is the sum of the distnces from F1 to P nd from F2 to P (your solution to 3c)? F1 b F2 b. Given tht the tringle is isosceles, wht is the distnce from F1 to point P? c. Use the Pythgoren Theorem to write n expression for the distnce from the origin to point F2. d. In your Geometry Expressions drwing, constrin the distnce from F1 to the origin to your nswer to 5c. Do the sme for the distnce from F2 to the origin. (You will need to drw point t the origin first). NOTE: If you wnt to type squre root in to Geometry Expressions, use sqrt, nd if you wnt to type in n exponent, use ^. For exmple, + b cn be typed: sqrt(^2 + b^2) e. Does P pper to fll on the ellipse? Drg it round. Does it sty on the ellipse? 2008 Sltire Softwre Incorported
8 6. If the curve in your Geometry Expressions drwing is truly n ellipse, then its implicit eqution will mtch the locus of point P. Select the curve, nd click on the Clculte Implicit Eqution icon. Now, hide the curve. Select the curve. Right click on the curve. Choose hide. Crete the locus of point P with respect to d, nd clculte its implicit eqution. Restore the originl curve by clicking on Show All in the View menu. Are the two implicit equtions the sme? How do they differ, if t ll? 7. How does trnsltion ffect the eqution of n ellipse? Open new drwing nd use Drw Function to crete new ellipse. X = cos( T ) Choose Prmetric for the type nd enter. Y = bsin( T ) u0 Crete vector, nd constrin it to its defult vlues, v 0. Trnslte the ellipse. Select the ellipse Click on Construct Trnsltion Click on the vector. Clculte the prmetric eqution of the new ellipse, nd record it in the box. Generl Prmetric Eqution of n Ellipse Clculte the implicit eqution of the new ellipse Sltire Softwre Incorported
9 ( x u ) ( ) 0 y v0 8. The Generl form for the implicit eqution of n ellipse is tht your implicit eqution is equivlent to the generl form. + =. Verify b 1 9. In prt 6, you found reltionship known s The Pythgoren Property for Ellipses 2 = b + c is hlf the horizontl xis of the ellipse b is hlf the verticl xis of the ellipse c is the distnce from the center of the ellipse to ech focus Digrm 4 b. Wht hppens if = b? b. Is it possible for < b? How would you need to modify the Pythgoren Property for the ellipse in Digrm 4? In ny ellipse, the lrger of 2 nd 2b is clled the mjor xis. The smller of 2 nd 2b is clled the minor xis. If > b, then the foci lie on horizontl line. If < b, then the foci lie on verticl line. If = b, then the ellipse is ctully circle Sltire Softwre Incorported
10 10. Summry: The generl prmetric form of the eqution of n ellipse is: where is the center of the ellipse, is the horizontl rdius of the ellipse, nd nd re the rdii of the ellipse. The generl implicit form of the eqution of n ellipse is: where is the center of the ellipse. If, then is the mjor dimeter nd is the minor dimeter. If, then is the mjor dimeter nd is the minor dimeter. The sum of the distnces from ny point on the ellipse to the two foci is. The distnce from the center of the ellipse to either focus follows the eqution. (, ) 2008 Sltire Softwre Incorported
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