8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1

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1 8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time. The LORAN equipment detects the difference in the rrivl times of the signls nd uses the locus definition of the hperol to determine the ship s loction. To determine the equtions of hperols, the solute vlues of numers re used. The solute vlue of rel numer is its distnce from zero on rel numer line. For rel numer represented, the solute vlue is written, which mens the positive vlue, or mgnitude, of. The digrm shows tht the solute vlue of 3 is 3 nd the solute vlue of 3 is = 3 3 = units 3 units 1 3 We Connection To lern more out LORAN, visit the ove we site. Go to Mth Resources, then to MATHEMATICS 11, to find out where to go net. Write rief report out the origins of LORAN. A hperol is the set or locus of points P in the plne such tht the solute vlue of the difference of the distnces from P to two fied points nd is constnt. P P =k The two fied points, nd, re clled the foci of the hperol. The line segments P nd P re clled the focl rdii of the hperol. 8. The Hperol MHR 37

2 INVESTIGATE & INQUIRE You will need two cler plstic rulers, sheet of pper, nd pencil. Step 1 Drw 10-cm line segment ner the centre of the piece of pper. Lel the two endpoints nd. Step Choose length, k centimetres, which is less thn the length. For this investigtion, use k = cm. Step 3 Choose pir of lengths, centimetres nd centimetres, such tht the solute vlue of their difference equls, tht is, =. For emple, choose = 9 cm nd = 5 cm, or = 5 cm nd = 9 cm, since 9 5 = nd 5 9 =. Step Use oth rulers to mrk two points tht re 9 cm from nd 5 cm from. Then, use oth rulers to mrk two points tht re 5 cm from nd 9 cm from. 9 cm 5 cm 5 cm 9 cm 9 cm 5 cm 5 cm 9 cm Step 5 Repet steps 3 nd using different vlues of nd, such tht =, until ou hve mrked enough points to define two curves. Step Drw smooth curve through ech set of points. The two curves form hperol. 1. How mn es of smmetr does the hperol hve?. In reltion to nd, where is the point of intersection of the es of smmetr? 3. In Step, ou mrked four points for the chosen vlues of nd. Are there n vlues of nd for which onl two points cn e mrked? If so, wht re the vlues of nd? Descrie the loctions of the points on the hperol.. The vertices of this hperol re locted on. How is the distnce etween the vertices of the two curves relted to the vlue of k? Eplin. 38 MHR Chpter 8

3 EXAMPLE 1 Finding the Eqution of Hperol From its Locus Definition Use the locus definition of the hperol to find n eqution of the hperol with foci ( 5, 0) nd (5, 0), nd with the constnt difference etween the focl rdii equl to 8. SOLUTION Let P(, ) e n point on the hperol. The locus definition of the hperol cn e stted lgericll s P P = 8. Use the formul for the length of line segment, l = ( 1 ) + (, 1 ) to rewrite P nd P. l = ( 1 ) + ( 1 ) P = ( 5)) ( + ( 0) = ( + 5) + P = ( 5) + ( 0) = ( 5) + Without loss in generlit, we ssume tht P > P nd sustitute. Sustitute: ( + 5) + ( 5) + = 8 Isolte rdicl: ( + 5) + = 8 + ( 5) + Squre oth sides: ( + 5) + = + 1 ( 5) + + ( 5) = + 1 ( 5) Isolte the rdicl: 0 = 1 ( 5) + Divide oth sides : 5 1 = ( 5) + Squre oth sides: = 1(( 5) + ) Simplif: = 1( ) = = 1 Divide oth sides 1: = An eqution of the hperol is = The Hperol MHR 39

4 The hperol in Emple 1 cn e modelled grphicll, s shown. The hperol hs two es of smmetr. The points (, 0) nd (, 0) lie on one is of smmetr. The line segment joining these two points is clled the trnsverse is. In this cse, the length of the trnsverse is is 8. The endpoints of the trnsverse is re the vertices of the hperol. 3 = The points (0, 3) nd (0, 3) lie on the other is of smmetr. The line segment joining these two points is clled the conjugte is. In this cse, the length of the conjugte is is. The endpoints of the conjugte is re clled the co-vertices of the hperol. The point of intersection of the trnsverse is nd the conjugte is is clled the centre of the hperol. In this cse, the centre is the origin, (0, 0). The lines = nd = form rectngle with the lines = 3 nd = 3. The grph of the hperol lies etween the digonls of this rectngle. The digonls of this rectngle re clled the smptotes of the hperol. These re the lines tht the hperol pproches for lrge vlues of nd. The eqution of the hperol from Emple 1 cn e written s = 1 3 (0, 3) ( 5, 0) (, 0) (, 0) (0, 3) 3 = (5, 0) In this form of the eqution, notice tht is hlf the length of the trnsverse is, or hlf the difference etween the focl rdii, nd 3 is hlf the length of the conjugte is. Notice lso tht the equtions of the smptotes re = 3 nd = 3. The coordintes of the foci re ( 5, 0) nd (5, 0). Notice tht + 3 = 5. 0 MHR Chpter 8

5 The stndrd form of the eqution of hperol centred t the origin, with the trnsverse is long the -is nd the conjugte is long the -is, is = 1 The length of trnsverse is long the -is is. The length of conjugte is long the -is is. The vertices re V 1 (, 0) nd V (, 0). The co-vertices re (0, ) nd (0, ). The foci re ( c, 0) nd (c, 0), which re on the trnsverse is. The equtions of the smptotes re = nd =. The stndrd form of the eqution of hperol centred t the origin, with the trnsverse is long the -is nd the conjugte is long the -is, is = 1 The length of trnsverse is long the -is is. The length of conjugte is long the -is is. The vertices re V 1 (0, ) nd V (0, ). The co-vertices re (, 0) nd (, 0). The foci re (0, c) nd (0, c), which re on the trnsverse is. The equtions of the smptotes re = nd =. + = c + = c (0, ) (0, c) ( c, 0) conjugte is trnsverse is V 0 1 (, 0) V (, 0) (0, ) (c, 0) is V (0, ) (, 0) conjugte is (, 0) 0 trnsverse V 1 (0, ) (0, c) 8. The Hperol MHR 1

6 EXAMPLE Sketching the Grph of Hperol With Centre (0, 0) Sketch the grph of the hperol = 1. Lel the foci nd the smptotes. 3 SOLUTION Since the eqution is in the form = 1, the centre is (0, 0) nd the trnsverse is is long the -is. Since = 3, =, nd the vertices re V 1 (0, ) nd V (0, ). Since =, =, nd the co-vertices re (, 0) nd (, 0). The length of the trnsverse is is = () = 1 The length of the conjugte is is = () = The foci re (0, c) nd (0, c). To find c, use + = c, where = nd =. c = + = + = 3 + = 0 c = 0 = 10 The coordintes of the foci re (0, 10 ) nd (0, 10 ), or pproimtel (0,.3) nd (0,.3). The equtions of the smptotes re = nd =. Sustituting for nd gives =, or = 3, nd =, or = 3. To sketch the grph of the hperol, plot the vertices nd the co-vertices. Use the lines =, =, =, nd = to construct rectngle. Sketch the smptotes etending the digonls of the rectngle. Since the trnsverse is is long the -is, the hperol must open up nd down. To sketch rnch of the hperol, strt t verte nd pproch the smptotes. Sketch the other rnch in the sme w. Lel the foci nd the smptotes. MHR Chpter 8 = 3 8 (0, 10) V (0, ) (, 0) (, 0) = 3 V 1 (0 1, ) (0, 10)

7 A hperol m not e centred t the origin. As in the cse of the circle nd the ellipse, hperol cn hve centre (h, k). The trnsltion rules tht ppl to the circle nd the ellipse lso ppl to the hperol. The stndrd form of the eqution of hperol centred t (h, k), with the trnsverse is prllel to the -is nd the conjugte is prllel to the -is, is ( h) ( k) = 1 The length of the trnsverse is prllel to the -is is. The length of the conjugte is prllel to the -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 (h + c, k), which re on the trnsverse is. + = c k + (h, k + ) The stndrd form of the eqution of hperol centred t (h, k), with the trnsverse is prllel to the -is, nd conjugte is prllel to the -is, is ( k) ( h) = 1 The length of the trnsverse is prllel to the -is is. The length of the conjugte is prllel to the -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 (h, k + c), which re on the trnsverse is. + = c k + c (h, k + c) conjugte is V (h +, k) (h c, k) k trnsverse (h, k) is (h + c, k) V 1 (h, k) 0 h c h h h + h + c k (h, k ) V k + (h, k + ) k 0 k c is k conjugte is (h +, k) (h, k) (h, k) trnsverse V 1 (h, k ) h h h + c (h, k c) 8. The Hperol MHR 3

8 EXAMPLE 3 Sketching the Grph of Hperol With Centre (h, k) Sketch the grph of the hperol ( ) ( + 1) = 1. Lel the foci. 3 1 SOLUTION The centre is C(h, k) = (, 1). Since the eqution is in the form ( h) ( k) = 1, the trnsverse is is prllel to the -is. = 3, so = = 1, so = The length of the trnsverse is is, or 1. The length of the conjugte is is, 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) nd V ( +, 1), or V 1 (, 1) nd V (8, 1). The co-vertices re (h, k ) nd (h, k + ). Sustitute the vlues of h, k, nd. The co-vertices re (, 1 ) nd (, 1 + ), or (, 5) nd (, 3). The foci re (h c, k) nd (h + c, k). To find c, use + = c, with = nd =. c = + = = 5 c = 5 = 13 The coordintes of the foci re ( 13, 1) nd ( + 13, 1), or pproimtel ( 5.1, 1) nd (9.1, 1). To sketch the grph of the hperol, plot the vertices nd the co-vertices. Then, construct the rectngle tht goes through ll four of these points. Sketch the smptotes etending the digonls of the rectngle. MHR Chpter 8

9 Since the trnsverse is is prllel to the -is, the hperol must open left nd right. To sketch rnch of the hperol, strt t verte nd pproch the smptotes. Sketch the other rnch in the sme w. Lel the foci. (, 3) ( 13, 1) V 1 (, 1) V (8, 1) ( + 13, 1) (, 5) EXAMPLE Writing n Eqution of Hperol Write n eqution for the hperol with vertices (, ) nd (, 8) nd co-vertices (0, 3) nd (, 3). Sketch the hperol. SOLUTION The vertices (, ) nd (, 8) lie on the trnsverse is, so the trnsverse is is prllel to the -is. The centre of the hperol is locted midw etween the vertices, nd is lso locted midw etween the co-vertices. Using the midpoint formul with the two vertices (, ) nd (, 8) gives, = = (, 3), + 8 Since the trnsverse is is prllel to the -is, nd the centre is not t the origin, the stndrd form of the eqution of the ellipse is ( k) ( h) = 1. (, 8) 8 (0, 3) (, 3) 0 (, ) Since the centre is (, 3), h = nd k = 3. The length of the trnsverse is, which is prllel to the -is, is 10. Since = 10, = 5. The length of the conjugte is, which is prllel to the -is, is. Since =, =. 8. The Hperol MHR 5

10 ( 3) The eqution of the hperol is ( ) = 1. 5 The foci re (h, k c) nd (h, k + c). To find c, use + = c, with = 5 nd =. c = + = 5 + = 5 + = 9 c = 9 The foci re locted t (h, k c) nd (h, k + c). Thus, the coordintes of the foci re (, 3 9 ) nd (, ), or pproimtel (,.39) nd (, 8.39). To sketch the grph of the hperol, plot the vertices nd the co-vertices. Then, construct the rectngle tht goes through ll four of these points. Sketch the smptotes etending the digonls of the rectngle. Since the trnsverse is is prllel to the -is, the hperol must open up nd down. To sketch rnch of the hperol, strt t verte nd pproch the smptotes. Sketch the other rnch in the sme w. Lel the foci (0, 3) (, 3 + 9) V (, 8) (, 3) 0 V 1 (, ) (, 3 9) Note tht hperols cn e grphed using grphing clcultor. As with circles nd ellipses, the equtions of hperols must first e solved for. For emple, solving = 1, results in = ± Enter oth of the resulting equtions in the Y= editor. Y1 = nd Y = MHR Chpter 8

11 Adjust the window vriles, if necessr, nd use the Zsqure instruction. Ke Concepts A hperol is the set, or locus, of points P in the plne such tht the solute vlue of the difference of the distnces from P to two fied points nd is constnt. P P = k The stndrd form of the eqution of hperol, with centre t the origin, is either = 1 (trnsverse is long the -is) or = 1 (trnsverse is long the -is). The stndrd form of the eqution of hperol, centred t (h, k), is either ( h) ( k) = 1 (trnsverse is prllel to the -is) ( k) or ( h) = 1 (trnsverse is prllel to the -is). Communicte Your Understnding 1. Descrie how ou would use the locus definition of the hperol to find n eqution of the hperol with foci (0, 5) nd (0, 5), nd with the constnt difference etween the focl rdii equl to 8.. Descrie how ou would sketch the grph of ( ) ( + ) = Descrie how ou would write n eqution in stndrd form for the hperol with vertices (3, ) nd (3, ) nd co-vertices (1, 1) nd (5, 1).. How would ou know if ) n eqution in stndrd form models n ellipse or hperol? ) grph models n ellipse or hperol? 5. Is hperol function? Eplin. 8. The Hperol MHR 7

12 Prctise A 1. Use the locus definition of the hperol to find n eqution for ech of the following hperols, centred t the origin. ) foci t ( 5, 0) nd (5, 0), with the difference etween the focl rdii ) foci t (0, 5) nd (0, 5), with the difference etween the focl rdii. For ech hperol, determine i) the coordintes of the centre ii) the coordintes of the vertices nd co-vertices iii) the lengths of the trnsverse nd conjugte es iv) n eqution in stndrd form v) the coordintes of the foci vi) the domin nd rnge ) c) d) 3. Sketch the grph of ech hperol. Lel the coordintes of the centre, the vertices, the co-vertices, nd the foci, nd the equtions of the smptotes ) = 1 ) = ) 0 c) 5 = 100 d) 9 = 3. Determine n eqution in stndrd form for ech of the following hperols. ) vertices (0, ) nd (0, ), co-vertices ( 5, 0) nd (5, 0) ) vertices ( 3, 0) nd (3, 0), foci ( 5, 0) nd (5, 0) c) foci ( 5, 0) nd (5, 0), with constnt difference etween focl rdii d) foci (0, 3) nd (0, 3), with constnt difference etween focl rdii 5 8 MHR Chpter 8

13 5. For ech hperol, determine i) the coordintes of the centre ii) the coordintes of the vertices nd co-vertices iii) the lengths of the trnsverse nd conjugte es iv) n eqution in stndrd form v) the coordintes of the foci ) d) 0 ) c) Sketch the grph of ech hperol. Lel the coordintes of the centre, the vertices, the co-vertices, nd the foci. ( 3) ( + 1) ) = ( + ) ) ( 1) = c) 1 9( ) = 1 d) ( ) 9( 5) = 3 7. Determine n eqution in stndrd form for ech of the following hperols. ) vertices (, 5) nd (, 5), foci (, 5) nd (8, 5) ) centre (0, 3), one verte (0, 5), nd one focus (0, ) c) centre ( 3, 1), one focus ( 5, 1), nd length of conjugte is d) centre (, ), one focus (, 5), nd length of conjugte is 8. The Hperol MHR 9

14 Appl, Solve, Communicte B 8. Mrine iolog A ship is monitoring the movement of pod of whles with its rdr. The rdr screen cn e modelled s coordinte grid with the ship t the centre (0, 0). The pod ppers to e moving long curve such tht the solute vlue of the difference of its distnces from (, 7) nd (, 3) is lws. Write n eqution in stndrd form to descrie the pth of the pod. 9. Motion in spce Some comets trvel on hperolic pths nd never return. Suppose the hperolic pth of comet is modelled on grid with scle of 1 unit = km. The verte of the pth is t (, 0), nd the focus is t (, 0). Write n eqution in stndrd form to model the pth of the comet. 10. Conjugte hperols ) Two hperols re centred t (, 1). One hs trnsverse is prllel to the -is, nd the other hs trnsverse is prllel to the -is. The shre the sme pir of smptotes. The eqution ( 1) of one hperol is ( ) = 1. Find the lengths of the 5 9 conjugte nd trnsverse es, nd the eqution in stndrd form, of the other hperol. ) The two hperols in prt ) re known s conjugte hperols. Write equtions in stndrd form for nother pir of conjugte hperols with their common centre not t the origin. Eplin our resoning. 11. Appliction Two prk rngers were sttioned t seprte loctions on the sme side of lke. The rngers were km prt. Both rngers sw lightning olt strike the ground on the other side of the lke nd herd the clp of thunder. ) If one rnger herd the clp of thunder s efore the other, write n eqution tht descries ll the possile loctions of the thunder clp. Plce the two rngers on the -is, with the midpoint etween the rngers t the origin. The speed of sound is pproimtel 0.35 km/s. ) Sketch the possile loctions where the lightning olt hit the ground. Include the rnger sttions in the sketch. 50 MHR Chpter 8

15 1. Technolog Use grphing clcultor to grph ech of the following. ( + ) ( 1) ( + 1) ) = 1 ( + 3) ) = Nvigtion The LORAN sstem uses the locus definition of the hperol. The sstem mesures the difference in the rrivl times t ship or ircrft of rdio signls from two sttions. The difference in the rrivl times is converted to difference in the distnces of the ship or ircrft from the two sttions. Two pirs of sttions sending rdio signls define two hperols. The ect loction of the ship or ircrft is point of intersection of the two hperols. ) On grid with scle in kilometres, plot nd lel the foci (, 0) nd (, 0) of hperol. The foci represent two sttions, nd, tht send rdio signls to ship positioned t point P. Find n eqution of the hperol, if the difference in the rrivl times of the rdio signls t P is converted to the difference in distnces P P = 8. ) On the sme grid s in prt ), plot nd lel the foci F 3 (0, 5) nd F (0, 5) of nother hperol. The foci represent nother two sttions, F 3 nd F, tht send rdio signls to the sme ship t point P. Find n eqution of this hperol, if F 3 P F P =. c) Sketch the two hperols nd estimte the coordintes of the points of intersection. How mn points of intersection re there? d) Use grphing clcultor to find the coordintes of the points of intersection of the two hperols, to the nerest tenth. If the second qudrnt of the grid is the onl qudrnt tht contins od of wter, wht re the coordintes of the ship? 1. Communiction ) Use grphing clcultor to grph the fmil of hperols = 1 for = 1,, nd 3. 5 ) Now, grph for = 1, 1 3, 1. c) How re the grphs like? How re the different? d) Wht hppens to the hperol s gets closer to 0? C 15. Inquir/Prolem Solving The eccentricit, e, of hperol is defined s e = c. Since c >, it follows tht e > 1. ) Descrie the generl shpe of hperol whose eccentricit is close to 1. ) Descrie the shpe if e is ver lrge. 8. The Hperol MHR 51

16 1. Equilterl hperols The following re emples of equilterl hperols. i) = ii) = 9 iii) = 1 iv) = 5 ) Wht do the equtions hve in common? ) Grph ech hperol. Stte the equtions of its smptotes. c) Wht do the grphs hve in common tht is different from the other hperols ou hve grphed? 17. Stndrd form Consider hperol with its trnsverse is prllel to the -is, foci t ( c, 0) nd (c, 0), nd vertices t (, 0) nd (, 0). ) Use the formul for the length of line segment, l = ( 1 ) + (, 1 ) to show tht, for point P(, ) on the hperol, ( c) + ( + c) + = nd ( + c) + ( c) + =. ) From the equtions in prt ), derive = 1. c c) Derive = 1, the stndrd form for the hperol centred t (0, 0) with trnsverse is prllel to the -is. 18. Inverse vrition When one vrile increses nd nother vrile decreses, such tht their product is constnt, the reltionship cn e represented = k, k 0. The reltionship cn e stted verll s vries inversel s. The grph of the inverse vrition is one rnch of hperol, for, > 0. Complete the following using grphing clcultor. ) Grph =, = 5, nd =10. Wht re the similrities nd differences for = k? ) Grph =, = 5, nd = 10. Compre these grphs with the grphs in prt ). c) Eplin wh the grphs in prts ) nd ) do not go through the origin. d) Cn ou grph = 0? Eplin. A CHIEVEMENT Check Knowledge/Understnding Thinking/Inquir/Prolem Solving Communiction Appliction Find n eqution for the locus of points such tht the product of the slopes of the lines from point on the locus to the points (, 0) nd (, 0) is. 5 MHR Chpter 8

, are called the foci (plural of focus) of the ellipse. The line segments F 1. P are called focal radii of the ellipse.

, are called the foci (plural of focus) of the ellipse. The line segments F 1. P are called focal radii of the ellipse. 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.