8.5 The Ellipse Kidne stones re crstl-like ojects tht cn form in the kidnes. Trditionll, people hve undergone surger to remove them. In process clled lithotrips, kidne stones cn now e removed without surger. To remove the stones, doctors cn use lithotripter, which mens stone crusher in Greek. A lithotripter is ellipticll-shped, nd its design mkes use of the properties of the ellipse to provide sfer method for removing kidne stones. This method will e eplined in Emple 5. An ellipse is the set or locus of points P in the plne such tht the sum of the distnces from P to two fied points nd F is constnt. P + F P = k We Connection www.school.mcgrwhill.c/resources/ To lern more out lithotripters, visit the ove we site. Go to Mth Resources, then to MATHEMATICS 11, to find out where to go net. Write rief report out how lithotripters work. The two fied points, nd F, re clled the foci (plurl of focus) of the ellipse. The line segments P nd F P re clled focl rdii of the ellipse. INVESTIGATE & INQUIRE You will need two cler plstic rulers, sheet of pper, nd pencil for this investigtion. Step 1 Drw 10-cm line segment ner the centre of the piece of pper. Lel the two endpoints nd F. F Step Choose length, k centimetres, which is greter thn the length F. You m wnt to mke k less thn 0 cm. Step 3 Choose pir of lengths, centimetres nd centimetres, where k = +. 8.5 The Ellipse MHR 619
Step 4 Use oth rulers to mrk two points tht re centimetres from nd centimetres from F. F Step 5 Repet steps 3 nd 4 using different vlues for nd until ou hve mrked enough points to define complete curve. Step 6 Drw smooth curve through the points. The curve is n emple of n ellipse. 1. How mn es of smmetr does the ellipse hve?. In reltion to nd F, where is the point of intersection of the es of smmetr? 3. ) Wh must k e greter thn the length of the line segment F? ) Wht digrm would result if k equlled F? 4. In step 4, ou locted two points for the chosen vlues of nd. Are there n vlues of nd for which onl one point cn e mrked? If so, wht re the vlues of nd? Where is the loction of the point on the ellipse? The digrm t the right shows nother method for drwing n ellipse. Pushpins t nd F re used to fsten loop of string, which is over twice s long s F. If ou hold pencil tight ginst the string, nd move the pencil long the string, ou will trce n ellipse. Since the length of string is constnt, the sum of the distnces P nd F P is constnt for ll positions of P. P string F 60 MHR Chpter 8
EXAMPLE 1 Finding the Eqution of n Ellipse From its Locus Definition Use the locus definition of the ellipse to find n eqution of n ellipse with foci ( 4, 0) nd F (4, 0), nd with the constnt sum of the focl rdii equl to 10. SOLUTION Let P(, ) e n point on the ellipse. The locus definition of the ellipse cn e stted lgericll s P + F P = 10. Use the formul for the length of line segment, l = ( 1 ) + (, 1 ) to rewrite P nd F P. l = ( 1 ) + ( 1 ) P = ( 4)) ( + ( 0) = ( + 4) + F P = ( 4) + ( 0) = ( 4) + Sustitute : ( + 4) + + ( 4) + = 10 Isolte rdicl: ( + 4) + = 10 ( 4) + Squre oth sides: ( + 4) + = 100 0 ( 4) + + ( 4) + Simplif: + 8 + 16 + = 100 0 ( 4) + + 8 + 16 + Isolte the rdicl: 16 100 = 0 ( 4) + Divide oth sides 4: 4 5 = 5 ( 4) + Squre oth sides: 16 00 + 65 = 5(( 4) + ) Simplif: 16 00 + 65 = 5( 4) + 5 16 00 + 65 = 5( 8 + 16) + 5 16 00 + 65 = 5 00 + 400 + 5 5 = 9 + 5 5 9 Divide ech side 5: 1 = + The eqution of the ellipse is + = 1. 5 9 8.5 The Ellipse MHR 61
An ellipse hs two es of smmetr. The longer line segment is clled the mjor is, nd the shorter line segment is clled the minor is. The endpoints on the mjor is re the vertices of the ellipse. The endpoints of the minor es re the co-vertices of the ellipse. verte co-verte minor is mjor is co-verte verte The ellipse in Emple 1 cn e modelled grphicll, s shown. Note tht the coordintes of the vertices re (5, 0) nd ( 5, 0). The length of the mjor is is 10. The coordintes of the co-vertices re (0, 3) nd (0, 3). The length of the minor is is 6. The eqution cn e written s + = 1. 5 3 In this form of the eqution, notice tht 5 is hlf the length of the mjor is, or hlf the constnt sum of the focl rdii, nd 3 is hlf the length of the minor is. In the digrm, hlf the length of the mjor is is denoted. Hlf the length of the minor is is denoted. Hlf the distnce etween the two foci, which re on the mjor is, is denoted c. ( 5, 0) 4 6 4 0 4 + =1 5 9 ( 4, 0) c 4 (0, 3) (0, 3) F (4, 0) (5, 0) If P is point on the ellipse when = 0, the focl rdii P = c nd F P = + c. P + F P = c + + c = So, the constnt sum of the distnces from point on the ellipse to the two foci is, which is the length of the mjor is. P + c F c 6 MHR Chpter 8
Let P e point on the ellipse when = 0. Since P + F P =, nd P = F P, then P = nd F P =. P Using the Pthgoren theorem, = + c, with >. Notice tht, in Emple 1, = 5, = 3, nd c = 4, nd tht 5 = 3 + 4. c 0 c F Therefore, the mjor is hs length of units, the minor is hs length of units, nd the distnce etween the two foci is c units. The following digrms show how the ke points of ellipses centred t the origin re lelled. Horizontl mjor is Co-verte (0, ) Verticl mjor is Verte (0, ) Verte (, 0) Minor is Mjor is 0 Focus ( c, 0) Focus F (c, 0) Co-verte (0, ) Verte (, 0) Co-verte (, 0) Minor Mjor is 0 Focus (0, c) Focus F (0, c) is Co-verte (, 0) Verte (0, ) 8.5 The Ellipse MHR 63
These results cn e summrized s follows. Ellipse with centre t the origin nd mjor is long the -is. The stndrd form of the eqution of n ellipse centred t the origin, with the mjor is long the -is is + = 1, > > 0 V 1 (, 0) Mjor is V (, 0) F 1 ( c, 0) 0 F (c, 0) The length of the mjor is is. The length of the minor is is. The vertices re V 1 (, 0) nd V (, 0). The co-vertices re (0, ) nd (0, ). The foci re ( c, 0) nd F (c, 0). = + c Minor is (0, ) (0, ) Ellipse with centre t the origin nd mjor is long the -is. The stndrd form of the eqution of n ellipse centred t the origin, with the mjor is long the -is is + = 1, > > 0 (, 0) Minor Mjor is V (0, ) 0 F (0, c) (0, c) is V 1 (0, ) (, 0) The length of the mjor is is. The length of the minor is is. The vertices re V 1 (0, ) nd V (0, ). The co-vertices re (, 0) nd (, 0). The foci re (0, c) nd F (0, c). = + c 64 MHR Chpter 8
EXAMPLE Sketching the Grph of n Ellipse With Centre (0, 0) Sketch the grph of the ellipse 4 + = 36. Lel the foci. SOLUTION Rewrite 4 + = 36 in stndrd form. 4 + = 36 Divide oth sides 36: + = 1 9 36 Since the denomintor of is less thn the denomintor of, the eqution is in the form + = 1. The ellipse is centred t the origin nd the mjor is is on the -is. Since = 36, = 6, nd the vertices re V 1 (0, 6) nd V (0, 6). Since = 9, = 3, nd the co-vertices re ( 3, 0) nd (3, 0). = + c 6 = 3 + c 36 9 = c 7 = c 3 3 = c The foci re (0, 3 3 ) nd F (0, 3 3 ), or V (0, 6) 6 pproimtel (0, 5.) nd (0, 5.). 4 + = 36 F (0, 3 3) 4 Plot the vertices nd co-vertices. Drw smooth curve through the points. Lel the foci nd the grph. ( 3, 0) (3, 0) 6 4 0 4 6 4 6 (0, 3 3) V 1 (0, 6) An ellipse m not e centred t the origin. As in the cse of the circle, n ellipse cn hve centre (h, k). The trnsltion rules tht ppl to the circle lso ppl to the ellipse. 8.5 The Ellipse MHR 65
The stndrd form of the eqution of n ellipse with centre (h, k) nd the mjor is prllel to the -is is ( h) ( k) + = 1, > > 0 k + (h, k + ) The stndrd form of the eqution of n ellipse with centre (h, k) nd the mjor is prllel to the -is is ( h) ( k) + = 1, > > 0 k + V (h, k + ) F (h, k + c) V 1 (h, k) k (h c, k) (h, k) F (h + c, k) V (h +, k) k (h, k) (h, k) (h +, k) k h 0 (h, k ) h + The length of the mjor is is. The length of the minor is is. The vertices re V 1 (h, k) nd V (h +, k). The co-vertices re (h, k ) nd (h, k + ). The foci re (h c, k) nd F (h + c, k). = + c h k 0 h h (h, k c) V 1 (h, k ) h + The length of the mjor is is. The length of the minor is is. The vertices re V 1 (h, k ) nd V (h, k + ). The co-vertices re (h, k) nd (h +, k). The foci re (h, k c) nd F (h, k + c). = + c EXAMPLE 3 Sketching the Grph of n Ellipse With Centre (h, k) Sketch the grph of the ellipse ( + 1) ( 3) + = 1. Lel the foci. 5 16 SOLUTION Since the denomintor of ( + 1) is greter thn the denomintor of ( 3), the eqution ( + 1) ( 3) + = 1 is in the form 5 16 ( h) ( k) + = 1. The ellipse is centred t (h, k), or ( 1, 3), nd the mjor is is prllel to the -is. 66 MHR Chpter 8
= 5, so = 5 = 16, so = 4 The mjor is, which is prllel to the -is, hs length of, or 10. The minor is, which is prllel to the -is, hs length of, or 8. The vertices re V 1 (h, k) nd V (h +, k). Sustitute the vlues of h, k, nd. The vertices re V 1 ( 1 5, 3) nd V ( 1 + 5, 3), or V 1 ( 6, 3) nd V (4, 3). The co-vertices re (h, k ) nd (h, k + ). Sustitute the vlues of h, k, nd. The co-vertices re ( 1, 3 4) nd ( 1, 3 + 4), or ( 1, 1) nd ( 1, 7). The foci re (h c, k) nd F (h + c, k). To find c, we use = + c, with = 5 nd = 4. = + c 5 = 3 + c 5 = 16 + c 5 16 = c 9 = c 3 = c The foci re ( 1 3, 3) nd F ( 1 + 3, 3), or ( 4, 3) nd F (, 3). Plot the vertices nd co-vertices. Drw smooth curve through the points. Lel the foci nd the grph. ( + 1) 5 + ( 3) 16 = 1 8 ( 1, 7) 4 ( 1, 3) V 1 ( 6, 3) V (4, 3) ( 4, 3) F (, 3) 6 10 8 6 4 0 4 6 ( 1, 1) EXAMPLE 4 Writing n Eqution of n Ellipse Write n eqution in stndrd form for ech ellipse ) The coordintes of the centre re (, 1). The mjor is hs length of 16 units nd is prllel to the -is. The minor is hs length of 4 units. ) The coordintes of the centre re (, 5). The ellipse psses through the points ( 5, 5), (1, 5), (, ), nd (, 1). 8.5 The Ellipse MHR 67
SOLUTION ) The centre is (, 1), so h =, nd k = 1. The mjor is is prllel to the -is. The length of the mjor is is 16, so = 8. The length of the minor is is 4, so =. Sustitute known vlues into the generl formul for n ellipse whose mjor is is prllel to the -is. ( h) ( k) + = 1 ( ) 8 ( ( 1)) + = 1 ( ) + ( + 1) = 1 64 4 ( + 1) The eqution of the ellipse is ( ) + = 1. 64 4 ) Plot the points ( 5, 5), (1, 5), (, ), nd (, 1). Drw smooth curve through the points. Lel the centre (, 5). The centre is (, 5) so h = nd k = 5. From the sketch of the ellipse, the mjor is is prllel to the -is, nd is 14 units in length, so = 7. The minor is is 6 units in length, so = 3. Sustitute known vlues into the generl formul for n ellipse whose mjor is is prllel to the -is. ( h) ( ( )) 3 ( + ) 9 ( k) + = 1 ( 5) + = 1 7 ( 5) + = 1 49 ( + ) 9 + ( 5) 49 ( 5, 5) = 1 10 8 4 (, 1) 1 6 (, 5) 10 8 6 4 0 (, ) (1, 5) The eqution of the ellipse is ( + ) ( 5) + = 1. 9 49 68 MHR Chpter 8
Note tht ellipses cn e grphed using grphing clcultor. As with circles, the equtions of ellipses must first e solved for. ( + 1) ( ) + = 1 9 4 ( + 1) ( ) 36 + 36 = 36 1 9 4 4( + 1) + 9( ) = 36 9( ) = 36 4( + 1) ( ) = 1 9 (36 4( + 1) ) = ± 1 9 (36 4( + 1) ) = ± 1 3 36 ( 4 + 1) Then, enter oth of the resulting equtions in the Y= editor. Y1 = + 1 3 36 ( 4 + 1) nd Y = 1 3 36 ( 4 + 1) Adjust the window vriles nd use the Zsqure instruction. The window vriles include Xmin = 5.3, Xm = 5.3, Ymin =, nd Ym = 5. EXAMPLE 5 Lithotrips Lithotrips is eing used to provide n lterntive method for removing kidne stones. A kidne stone is crefull positioned t one focus point of the ellipticll shped lithotripter. Shock wves re sent from the other focus point. The reflective properties of the ellipse cuse the shock wves to intensif, destroing the kidne stone locted t the focus. Suppose tht the length of the mjor is of lithotripter is 60 cm, nd hlf of the length of the minor is is 5 cm. F Kidne Stone 8.5 The Ellipse MHR 69
) Write n eqution of the semi-ellipse. Assume tht the centre is t the origin nd tht the mjor is is long the -is. ) How fr must the kidne stone e from the source of the shock wves, to the nerest tenth of centimetre? SOLUTION ) Drw digrm. The ellipse is centred t the origin with mjor is long the -is, so the eqution is of the form + = 1 The length of the mjor is is 60, so = 30. Hlf the length of the minor is is 5, so = 5. An eqution of the semi-ellipse is + = 1, 0 30 5 0 60 cm F 5 cm or + = 1, 0 900 65 ) Find the coordintes of the foci. The foci re ( c, 0) nd F (c, 0). = + c 30 = 5 + c 900 65 = c 75 = c 75 = c 5 11 = c The foci re ( 5 11, 0) nd F (5 11, 0). The distnce etween the two foci is 10 11 = 33. The kidne stone must e 33. cm from the source of the shock wves, to the nerest tenth of centimetre. 630 MHR Chpter 8
Ke Concepts An ellipse is the set or locus of points P in the plne such tht the sum of the distnces from P to two fied points nd F is constnt. The stndrd form of the eqution of n ellipse, centred t the origin, with > > 0, is either + = 1 (mjor is on the -is) or + = 1 (mjor is on the -is). The stndrd form of the eqution of the ellipse, with centre (h, k) nd with > > 0, is either ( h) ( k) + = 1 (mjor is prllel to the ( k) -is) or ( h) + = 1 (mjor is prllel to the -is). Communicte Your Understnding 1. If the ellipses ( + 3) ( ) + = 1 nd ( + 3) ( ) + = 1 4 9 9 4 were grphed, wht fetures would e the sme? Wht fetures would e different? Eplin.. Stte whether ech of the following sttements is lws true, sometimes true, or never true for n ellipse. Eplin our resoning. ) The length of the mjor is is greter thn the length of the minor is. ) The ellipse is function. c) The ellipse hs infinitel mn es of smmetr. d) For the ellipse, + = 1, < 0 nd < 0. 3. Descrie how ou would use the locus definition of the ellipse to find n eqution of n ellipse with centre (0, 0), foci (0, 4) nd F (0, 4), nd with the sum of the focl rdii equl to 10. 4. Descrie how ou would sketch the grph of ( + ) ( + 4) + = 1. 9 16 8.5 The Ellipse MHR 631
Prctise A 1. Use the locus definition of the ellipse to write n eqution in stndrd form for ech ellipse. ) foci ( 3, 0) nd (3, 0), with sum of focl rdii 10 ) foci (0, 3) nd (0, 3), with sum of focl rdii 10 c) foci ( 8, 0) nd (8, 0), with sum of focl rdii 0 d) foci (0, 8) nd (0, 8), with sum of focl rdii 0. Sketch the grph of ech ellipse. Lel the coordintes of the centre, the vertices, the covertices, nd the foci. ) + = 1 ) + = 1 5 16 4 36 c) + 16 = 64 d) 4 + = 36 e) 5 + 9 = 5 3. For ech of the following ellipses, i) find the coordintes of the centre ii) find the lengths of the mjor nd minor es iii) find the coordintes of the vertices nd co-vertices iv) find the coordintes of the foci v) find the domin nd rnge vi) write n eqution in stndrd form ) ) 4 0 4 0 4. Write n eqution in stndrd form for ech ellipse with centre (0, 0). ) The mjor is is on -is, the length of the mjor is is 14, nd the length of the minor is is 6. ) The length of the minor is is 6, nd the coordintes of one verte re ( 5, 0). c) The length of the mjor is is 1, nd the coordintes one co-verte re (5, 0). d) The coordintes of one verte re ( 8, 0), nd the coordintes of one focus re ( 55, 0). e) The coordintes of one focus re (0, 10 ), nd the length of the minor is is 6. 5. Sketch the grph of ech ellipse finding the coordintes of the centre, the lengths of the mjor nd minor es, nd the coordintes of the foci, the vertices, nd the co-vertices. ( + ) ( 3) ) + = 1 5 9 ( 3) ( + 1) ) + = 1 49 81 c) ( + 1) + 9( 3) = 36 d) 16( 3) + ( + ) = 16 6. For ech of the following ellipses, i) find the coordintes of the centre ii) find the lengths of the mjor nd minor es iii) find the coordintes of the vertices nd co-vertices iv) find the coordintes of the foci v) find the domin nd rnge vi) write n eqution in stndrd form 63 MHR Chpter 8
) ) 0 c) d) 0 4 4 6 8 4 6 4 0 0 4 6 4 6 8 7. Write n eqution in stndrd form for ech ellipse. ) centre (, 3), mjor is of length 1, minor is of length 4 ) centre (3, ) nd pssing through ( 4, ), (10, ), (3, 1), nd (3, 5) c) centre ( 1, ) nd pssing through ( 5, ), (3, ), ( 1, 4), nd ( 1, 8) d) foci t (0, 0) nd (0, 8), nd sum of focl rdii 10 e) foci t ( 1, 1) nd (9, 1), nd sum of focl rdii 6 Appl, Solve, Communicte B 8. Whisper Chmer Sttur Hll, locted in the United Sttes Cpitol, hs ellipticl wlls. Becuse of the reflective propert of the ellipse, the hll is known s the Whisper Chmer. ) President John Quinc Adms desk ws locted t one of the focus points, nd he ws le to listen in on mn privte converstions. Where would the converstions hve to tke plce for Adms to her them? Eplin. ) Write n eqution of the ellipse tht models the shpe of Sttur Hll. Assume tht the length of the mjor is is 10 m nd the length of the semi-minor is (hlf the minor is) is 0 m. 9. Kepler s First Lw Johnnes Kepler ws phsicist who devised the three lws of plnetr motion. Kepler s First Lw sttes tht ll plnets orit the sun in ellipticl pths, with the centre of the sun t one focus. The distnce from the sun to plnet continull chnges. The Erth is closest 0 m F 10 m 8.5 The Ellipse MHR 633
to the sun in Jnur. The closest point, or perihelion, is 1.47 10 8 km from the sun. The Erth is frthest from the sun in Jul. The frthest point, or phelion, is 1.5 10 8 km from the sun. Write n eqution of the ellipse tht models the Erth s orit out the sun. Assume tht the centre of the ellipse is t the origin nd tht the mjor is is long the -is. Perihelion 10. Motion in spce Hlle s Comet orits the sun out ever 76 ers. The comet trvels in n ellipticl pth, with the sun t one of the foci. At the closest point, or perihelion, the distnce of the comet to the sun is 8.8 10 7 km. At the furthest point, or phelion, the distnce of the comet from the sun is 5.3 10 9 km. Write n eqution of the ellipse tht models the pth of Hlle s Comet. Assume tht the sun is on the -is. 11. Appliction The Erth s moon orits the Erth in n ellipticl pth. The perigee, the point where the moon is closest to the Erth, is pproimtel 363 000 km from the Erth. The pogee, the point where the moon is furthest from the Erth, is pproimtel 405 000 km from the Erth. The Erth is locted t one focus. Write n eqution of the ellipse tht models the moon s orit out the Erth. Assume tht the Erth is on the -is. 1. Show tht the eqution of the of the circle + = 49 cn e written in the stndrd form of n eqution of n ellipse. 13. Sputnik I The first rtificil Erth-oriting stellite ws Sputnik I, lunched into n ellipticl orit the USSR in 1957. If this orit is modelled with the centre of the ellipse t the origin nd the mjor is long the -is, then the length of the mjor is is 1180 km, nd the length of the minor is is 935 km. The Erth is t one focus. ) Write n eqution of the ellipse tht models the orit of the stellite. ) Wht is the closest distnce of the stellite to the Erth? c) Wht is the furthest distnce of the stellite from the Erth? 14. Coin set The twelve Hopes nd Aspirtions Cndin millennium qurters cn e purchsed in n ellipticll shped souvenir set. ) If the ellipse hs vertices (0, 7) nd (30.94, 7), nd co-vertices (15.47, 0) nd (15.47, 14), write n eqution of the ellipse. ) Wht is the length of the mjor is? Sun Erth Aphelion 634 MHR Chpter 8
c) Wht is the length of the minor is? d) Sketch scle digrm of the souvenir set. 15. Spotlight When spotlight shines on stge, the spotlight illumintes n re in the shpe of n ellipse. Assume tht one focus of the ellipse is ( 1, 5), nd the sum of the focl rdii is 6. Write the eqution of the ellipse if the mjor is is prllel to the -is. 16. Jupiter Like ll plnets, Jupiter hs n ellipticl orit, with the centre of the sun locted t focus. The digrm gives the pproimte minimum nd mimum distnces from Jupiter to the sun, in millions of kilometres. Write n eqution of the ellipse tht models Jupiter s orit round the sun. Assume tht the centre of the sun is on the -is. 17. Covered entrnce A semi-ellipticl covering is to e uilt over n 8-m-wide rod nd the -m-wide sidewlks on either side of it tht led to n rts centre. If there is mimum clernce of 5 m over the rod, wht will e the minimum clernce over the rod, to the nerest hundredth of metre? 18. Technolog Use grphing clcultor to grph ech ellipse. ( ) ( + 1) ) + = 1 ( + 3) ( + 1) ) + = 1 ( 3) c) + = 1 16 4 4 5 36 16 19. Communiction ) Use grphing clcultor to grph the fmil of ellipses + = 1 for = 1,, nd 3. 5 ) Grph the fmil for = 1, 1 3, 1 4. c) How re the grphs like? How re the different? d) Wht hppens to the ellipses s gets closer to 0? 0. Distorted circle An ellipse cn e thought of s distorted circle. ) Sketch the grphs of the circle + = 1 nd the ellipse + = 1 5 9 on the sme grid. ) B wht fctor hs the circle epnded horizontll to form the ellipse in prt )? 740 810 Sun 5 m Jupiter m rod 8 m m 8.5 The Ellipse MHR 635
c) B wht fctor hs the circle epnded verticll to form the ellipse in prt )? d) Sketch the grphs of the circle + = 1 nd the ellipse + = 1 4 36 on the sme grid. e) B wht fctor hs the circle epnded horizontll to form the ellipse in prt d)? f) B wht fctor hs the circle epnded verticll to form the ellipse in prt d)? g) How cn ou recognize the horizontl nd verticl stretch fctors from the eqution of the ellipse? C 1. Eccentricit Ellipses cn e long nd nrrow, or nerl circulr. The mount of elongtion, or fltness, of n ellipse is mesured numer clled the eccentricit. To clculte the eccentricit, e, use the formul e = c. ) Clculte the eccentricit of ech of the following ellipses. Round nswers to the nerest hundredth. i) + = 1 ii) + = 1 iii) + = 1 iv) circle 9 4 16 5 36 ) Inquir/Prolem Solving Find the gretest nd lest possile eccentricities for n ellipse. Eplin our resoning.. Stndrd form Consider n ellipse with its mjor is long the -is, foci t ( c, 0) nd (c, 0), -intercepts t (, 0) nd (, 0), nd -intercepts t (0, ) nd (0, ). ) Use the distnce formul to show tht for point (, ) on the ellipse, ( c) + + ( + c) + =. ) Isolte one rdicl term in the eqution nd derive + = 1. c c) Derive + = 1, the stndrd form for the ellipse with its mjor is long the -is. A CHIEVEMENT Check Knowledge/Understnding Thinking/Inquir/Prolem Solving Communiction Appliction The roof of n ice ren is in the form of semi-ellipse. It is 100 m cross nd 1 m high. Wht is the length of stilizing em 5 m elow the roof? 636 MHR Chpter 8