TImth.com Algebr Constructing n Ellipse ID: 9980 Time required 60 minutes Activity Overview This ctivity introduces ellipses from geometric perspective. Two different methods for constructing n ellipse re presented nd explored. Students discover tht the sum of the distnces from point on n ellipse to its foci is lwys constnt. This fct is then used s the bsis for n lgebric derivtion of the generl eqution for n ellipse centered t the origin. Topic: Anlytic Geometry Conics & Polr Coordintes Derive the eqution (in rectngulr form) of n ellipse s the locus of point tht moves so tht its totl distnce from two fixed points ( f, 0) nd (f, 0) is constnt. Write the eqution of n ellipse with center t (0, 0) given its vertices nd co-vertices nd grph it. Techer Preprtion nd Notes This ctivity is pproprite for n Algebr or Preclculus clssroom. Students should hve experience using the distnce formul nd solving rdicl equtions. This ctivity is intended to be techer-led with students in smll groups. Notes for using the TI-Nspire Nvigtor System re included throughout the ctivity. The use of the Nvigtor System is not necessry for completion of this ctivity. To downlod the student TI-Nspire document (.tns file) nd student worksheet, go to eduction.ti.com/exchnge nd enter 9980 in the keyword serch box. Associted Mterils ConstructEllipse_Student.doc ConstructEllipse.tns Suggested Relted Activities To downlod ny ctivity listed, go to eduction.ti.com/exchnge nd enter the number in the keyword serch box. Properties of n Ellipse (TI-Nspire technology) 8430 Orbit Of Jupiter (TI-Nspire technology) 10035 Ellipse: Envelope of Lines (Cbri Jr.) (TI-84 Plus fmily) 791 Ellipse: Locus of Points (Cbri Jr.) (TI-84 Plus fmily) 790 NUMB3RS Seson Hrvest Wxing Ellipticl (TI-84 Plus fmily) 65 011 Texs Instruments Incorported Techer Pge Constructing n Ellipse
TImth.com Algebr Problem 1 Envelope construction Students should begin by exmining the circle on pge 1.3. They will see tht point F lies on dimeter of the circle, segment FP connects F to point P on the circle, nd perpendiculr line to segment FP through point P is drwn. Students re directed to drg point P round the circle nd then disply the locus (MENU > Construction > Locus) of the perpendiculr line s P trvels long the circle, which genertes the shpe of n ellipse. TI-Nspire Nvigtor Opportunity: Live Presenter See Note 1 t the end of this lesson. This method for constructing n ellipse is clled the envelope method. Explin to students tht the dimeter of the circle is equl to the width of the ellipse long its mjor xis (the longer of its two xes). The point F is specil type of fixed point tht cn be used to generte the ellipse. Tell students tht ellipses hve two such fixed points, clled foci (singulr: focus). Next, students will use the digrm on pge 1.3 to explore the other focus. They will need to first hide the locus of lines. Then students should reflect F over the y-xis using the Reflection tool from the Trnsformtion menu. Lbel the imge point F'. After constructing segment from F' to P nd line perpendiculr to segment F'P through point P, students should then drg point P nd notice tht this new line lso trces out n ellipse. The locus will confirm this, nd students should lso notice tht both ellipses trced out re identicl. By drgging the focus F (which in turn moves F ), students should find tht the loction of the foci ffects the shpe of the ellipse. TI-Nspire Nvigtor Opportunity: Screen Cpture See Note t the end of this lesson. 011 Texs Instruments Incorported Pge 1 Constructing n Ellipse
TImth.com Algebr Problem String nd pins construction An ellipse is defined s the set of points in plne such tht the sum of the distnces from two fixed points (foci) in tht plne is constnt. Students will now use this definition to construct n ellipse. Pge.3 contins segment with slider nd two dditionl points, f1 nd f, which will become the foci of the ellipse. The vlues of d1 nd d, determined by the slider, will be the distnces from f1 nd f (respectively) to the point on the ellipse. Students should clculte d3, which is the sum of d1 nd d. They will find tht the vlue of d3 remins constnt s they move the slider. Pge.4 directs students to construct two circles. After selecting the Compss tool from the Construction menu, students should click on the mesurement for the rdius, followed by the point to use s the center of the circle. Next, students should mrk the intersections of the circles nd then construct four segments (the rdii of the circles). Displying the loci of the ech intersection point s the slider trvels long the segment revels n ellipse. This construction is clled the string nd pins construction becuse it is trditionlly performed by wrpping piece of string (represented here by the segment nd slider) round two pins driven into flt surfce t the foci. TI-Nspire Nvigtor Opportunity: Screen Cpture See Note 3 t the end of this lesson. Problem 3 Semimjor nd semiminor xes Pge 3. shows n nimted point on n ellipse. Students re told tht s the point trvels round the curve, its distnce from the center of the ellipse chnges. They should determine tht the semiminor xis (minimum distnce) occurs when the point is on the y-xis t the co-vertices of the ellipse nd the semimjor xis (mximum distnce) occurs when the point is on the x-xis t the vertices of the ellipse. 011 Texs Instruments Incorported Pge Constructing n Ellipse
TImth.com Algebr Problem 4 Deriving the eqution of n ellipse Pge 4. shows n ellipse centered t the origin. The lengths of the segments connecting (x, y) to the foci re d1 nd d. Since the foci re equidistnt from the center of the ellipse, the coordintes of the foci re (c, 0) nd ( c, 0) Let be the distnce from the center of the ellipse to the vertex (, 0). Students will follow the steps on pges 4.3 4.7 to derive generl eqution for such n ellipse. 1. First, students drg (x, y) so tht y = 0. Since the distnce from (, 0) to (, 0) is equl to the sum of (, 0) to ( c, 0) plus the distnce from ( c, 0) to (, 0), the eqution is = d1 + d.. Now students will drg (x, y) to ny point on the ellipse. Using the distnce formul, students should write the following for d1 nd d. d1 ( x c) ( y 0) ( x c) y d ( x ( c)) ( y 0) ( x c) y 3. Substitute these into the eqution = d1 + d. ( x c) y ( x c) y 4. Simplify the rdicl expression by isolting rdicl on one side nd then squring both sides. ( x c) y ( x c) y ( x c) y ( x c) y ( x c) y 4 4 ( x c) y ( x c) y Isolte the remining rdicl on the left side nd simplify the right side. 4 ( x c) y 4 ( x c) y ( x c) y 1 x c y x xc c x xc c 4 1 ( x c) y ( 4 4 xc) 4 xc ( x c) y ( ) ( 4 ( ) ( )) 011 Texs Instruments Incorported Pge 3 Constructing n Ellipse
Squre both sides gin. TImth.com Algebr xc ( x c) y c ( x c) y xc x c x xc c y xc x c x c y x 5. Fctor out x, then simplify. c x x y c c x 1 y c x c y c x c y c 6. Divide both sides by c. c x y c x y 1 c 7. Students will now return to the digrm nd drg (x, y) so tht x = 0. This forms two right tringles with two congruent sides, i.e., two congruent tringles. Since the tringles re congruent, students cn conclude tht d1 = d. 8. Recll tht = d1 + d. Therefore = d1 + d1, nd = d1. 9. Define b s the distnce from the center of the ellipse to this point. Students cn use the Pythgoren Theorem to write n expression for b in terms of (which equls d1) nd c. b c Note: Students my think c b, but in this cse the hypotenuse hs length nd the sides hve lengths b nd c, producing the eqution b c. x y 10. So we cn write the eqution in Step 6 s 1. This is the generl eqution for b n ellipse centered t the origin with semimjor xis nd semiminor xis b. 011 Texs Instruments Incorported Pge 4 Constructing n Ellipse
TI-Nspire Nvigtor Opportunities TImth.com Algebr Note 1 Problem 1, Live Presenter Use Live Presenter to guide students through pges 1. to 1.5. Use the displyed hndheld screen to guide the discussion through these pges. Note Problem 1, Live Presenter Use Live Presenter to help students describe the reltionship between the foci nd the resulting loci. If there is time, sk students wht the loction is of the foci when the loci results in : circle ellipse hyperbol Note 3 Problem, Screen Cpture Use Live Presenter to monitor student progress s they work through the construction on pge.3, offering help s needed. 011 Texs Instruments Incorported Pge 5 Constructing n Ellipse