Review of conic sections Conic sections re grphs of the form REVIEW OF CONIC SECTIONS prols ellipses hperols P(, ) F(, p) O p =_p REVIEW OF CONIC SECTIONS In this section we give geometric definitions of prols, ellipses, nd hperols nd derive their stndrd equtions. The re If clled we write conic sections, 4p, then or conics, the stndrd ecuse eqution the result from intersecting cone with It plne opens s upwrd shown in if Figure p P(, ) nd. downwrd if p [s grph is smmetric with respect to the -is ecuse P. F (_c, ) F (c, ) F F =_p (, p) ( p, ) ellipse prol hperol (, p) =_p =_p The re clled conic sections ecuse the cn e found tking douled cone nd slicing it with plne: () =4p, p> () =4p, p< (c) =4p, p> FIGURE Conics FIGURE 4 ellipse prol hperol If we interchnge nd in (), we otin Thomson Brooks-Cole copright 7 5 _, is prol focus F Review of conic sections verte FIGURE F(, p) O FIGURE 3 is focus Conic vertesections directri re grphs of the form FIGURE FIGURE 3 directri P(, ) =_p prols ellipses hperols 5 F(, p) p O F prol 4p PARABOLAS += 4 REVIEW which OF is CONIC n SECTIONS eqution of the prol with focus p, A prol is the set of points in plne tht re equidistnt from fied point F (clled nd mounts to reflecting out the digonl line the focus) nd fied line (clled the right directri). if p This nd definition to the left is if illustrted p Figure. Eercise [see 59). Figure This 4, pr Notice tht the point hlfw etween grph the focus is smmetric nd the directri with respect lies on the prol; it is tor the with -is, ellipticl which croi clled the verte. The line through the focus perpendiculr to the directri is clled the High-intensit sound is of the prol. EXAMPLE Find the focus nd directri of the prol In the 6th centur = 5 destro it without dm Glileo showed the tht grph. the pth of projectile tht is shot into the nd recovers within ir t n ngle to the ground is prol. Since then, prolic shpes hve een used in designing utomoile hedlights, SOLUTION reflecting If we telescopes, write the eqution nd suspension s ridges. nd c (See Chllenge Prolem.4 for the reflection see propert tht 4p of prols, so p tht mkes 5. Thus HYPERBOLAS them focus so useful.) is p, We otin prticulrl simple eqution is for. The prol sketch is if shown we plce in Figure its verte 5. t the origin O nd its directri prllel to the -is s in Figure 3. If the focus A is hperol the point is, the p, s then the directri hs the eqution ELLIPSES p. If P, is n P(, point ) on the fied prol, points then F nd thef distnce from P to the focus is Hperols occur P An ellipse is the set of points in economics plne the sum (Bole s of wh F (_c, PF ) F nd s is constnt p F F (c, ) (see Figure cnt 6). ppliction These two fied of h focus). One of Kepler s lws is tht I nd the II orits (see of Eercise the pl nd F the distnce from F P to the directri with is the Sun pt. one (Figure focus. 3 illustrtes Notice the cse tht where the de p.) The defining propert of prol In order is tht to these otin distnces the simplest re is eqution equl: tht the for sum n of ellips dis the points c, s p nd p c, s in Figure 7 so tht the FIGURE 6 FIGURE the eqution of hp Let the sum of the distnces from point on the ellipse P is on the hperol when s Eercise 5 to sho is point on the ellipse when We get n equivlent eqution PF - PF = squring nd simplifing: distnces is PF P(, ) (See Chllenge Prolem.4 for FIGURE the reflection 5 propert of prols tht mkes them so useful.) P(, ) p =_p All conic sections stisf n eqution of the form p p p A + B + C + D + E + F = for some constnts A, B, C, D, E, F. Thomson Brooks-Cole copright 7 on Brooks-Cole copright 7 PF PF tht is, s c s c F (_c, ) F (c, ) 6 p p p p or s c s 4p where c Squring oth sides, we hve, re the vert FIGURE 7 An eqution of the prol with focus, c p nd cdirectri 4 p 4s is,which c is =_ = with respect to oth which simplifies 4p to s To nlze c the h (_, ) (, ) We squre gin: (_c, ) (c, ) c c 4 which ecomes c
Prols is focus F prol verte directri Prols hve focus F ( point) nd directri ( line), nd re defined s ll the points tht re equidistnt from the focus nd the directri. The lso hve verte (the point on the curve closest to the directri) nd n is (the line perpendiculr to the directri through the verte nd the focus). Ellipses P F F Ellipses hve two foci F nd F (oth points), nd re defined s ll the points (, ) such tht the distnce from F to (, ) plus the distnce from (, ) to F is fied. The lso hve mjor is (the line through the foci) nd vertices (the points on the curve intersecting the mjor is).
Hperols =_ = (_, ) (, ) (_c, ) (c, ) Hperols lso hve two foci F nd F (oth points), nd re defined s ll the points (, ) such tht the di erence etween the distnces from F to (, ) nd from (, ) to F is fied. The lso hve vertices (one point on ech piece which re closest to ech other) nd smptotes (lines which the curves pproch t infinit). Clculting equtions for conic sections: Prols P(, ) F(, p) O p =_p For prol with verte t the origin nd directri = p prllel to the -is, the definition ll the points P (, ) tht re equidistnt from the focus nd the directri ss first tht the focus hd etter e t (,p) (since the origin is on the curve), nd then tht + p = PF = p ( ) +( p). Solving for, this gives =4p. Similrl, if the prol hd verte t the origin nd directri = p, thenthefocusist (p, ) nd the eqution for the prol is =4p.
(, p) =_p =_p (, p) =4p, p > =4p, p < ( p, ) (p, ) =_p =_p =4p, p > =4p, p < Emples of prol prolems. Wht conic section is the curve =?Whtrell relevnt points nd lines? Ans. This is prol with verte (, ), focus(,p) nd directri = p where =4p (since is the squred vrile). Thus p = /4 =5/.. Wht conic section is the curve ( ) = ( + )? Ans. Shift the coordintes! Let ŷ = (so =ŷ +)nd ˆ = +(so =ˆ ). Then ŷ = (ˆ) is the prol in prt, nd we just hve to shift ll of our coordintes ck. verte: (ˆ, ŷ) =(, ), so (, ) =(, + ) = (, ) focus: (ˆ, ŷ) =(, 5/), so (, ) =(, 5/ + ) = (, 7/) directri: ŷ = 5/, so = 5/+= 3/
Emples of prol prolems. Wht conic section is the curve ( ) = ( + )? Ans. verte: (ˆ, ŷ) =(, ), so (, ) =(, + ) = (, ) focus: (ˆ, ŷ) =(, 5/), so (, ) =(, 5/ + ) = (, 7/) directri: ŷ = 5/, so = 5/+= 3/ 3. Wht conic section is the curve 4 4 =? Ans. Tr to put in form we recognize! Del with the nd stu seprtel nd complete whtever squres pper. 4 = ( +4) = ( +4 +4 4) So = (( + ) 4) = ( + ) +4; = + 4 4 = + ( + ) +4 4 = ( ) ( + ). Sme s prt! Doing prol prolems Look for or, ut not oth! Get into the proper form! (ˆ) =4pŷ or (ŷ) =4pˆ. If ou re given something like 4 4 =, seprtethe stu nd the stu nd complete squres, nd del with shifting coordintes if necessr.
8 Find the verte, focus, nd directri of the prol nd sketch its grph... 4 3. 4 4. 5. 8 3 6. 5 7. 8. 5 6 9 Find n eqution of the prol. Then find the focus nd directri. 9.. _
Clculting equtions for conic sections: Ellipses P(, ) F (_c, ) F (c, ) (_, ) (_c, ) (, ) (, ) c (c, ) (, _) For n ellipse with the foci on the -is t the points F =( c, ) nd F =(c, ) nd the sum of the distnces from point on the ellipse to the foci e >. Then for n point P = P (, ) on the curve = PF + PF = p ( + c) + + p ( c) +. Since >c,lete defined = c. Mnipulting the eqution ove gives + =. Clculting equtions for conic sections: Ellipses P(, ) F (_c, ) F (c, ) (_, ) (_c, ) (, ) (, ) c (c, ) (, _) For n ellipse with the foci on the -is t the points F =( c, ) nd F =(c, ) nd the sum of the distnces from point on the ellipse to the foci e >, + =, where = c. The mjor is is the -is nd the vertices re (±, ). Similrl, if the foci re the points F =(, c) nd F =(,c) nd the sum of the distnces from point on the ellipse to the foci e >, + =, where = c. The mjor is is the -is nd the vertices re (, ±).
Doing ellipse prolems Look for oth nd, with the sme sign. Get into the proper form! ˆ + For emple, if ou re given ŷ =. 9 + 6 = 44, divide oth sides 44 first nd fctor into squres: = 9 44 + 6 44 = 44/9 + 44/6 = 6 + + 9 =. 4 3 If ou re given something like + +8 = seprte the stu nd the stu nd complete squres, nd del with shifting coordintes if necessr.
6 Find the vertices nd foci of the ellipse nd sketch its grph... 9 5 3. 4 6 4. 64 4 5 5 5. 6. 9 8 4 7 6 4 7 7 8 Find n eqution of the ellipse. Then find its foci. 7. 8.
Clculting equtions for conic sections: Hperols P(, ) F (_c, ) F (c, ) For hperol with the foci on the -is t the points F =( c, ) nd F =(c, ) nd the di erence of the distnces from point on the ellipse to the foci e ± ( >). Then for n point P = P (, ) on the curve ± = PF PF = p ( + c) + p ( c) +. Let e defined c = +. Mnipulting the eqution ove gives =. Clculting equtions for conic sections: Hperols P(, ) (_, ) =_ = (, ) F (_c, ) F (c, ) (_c, ) (c, ) For hperol with the foci on the -is t the points F =( c, ) nd F =(c, ) nd the di erence of the distnces from point on the ellipse to the foci e ± ( >), =, where c = +. Note tht the -intercepts re ± (set =nd solve). There is no -intercept ( =hs no solutions). But s! ±, /! ±/. So the vertices re(±, ) nd the smptotes re = ±(/).
Clculting equtions for conic sections: Hperols =_ (, c) = (, ) (, _) (, _c) Similrl, for hperol with the foci on the -is t the points F =(, c) nd F =(,c) nd the di erence of the distnces from point on the ellipse to the foci e ± ( >), =, where c = +. The -intercepts re ± (set =nd solve). There is no -intercept ( =hs no solutions). And s! ±, /! ±/. Sotheverticesre(, ±) nd the smptotes re = ±(/). (Switchll s for s.) Doing hperol prolems Look for oth nd,withdi erent signs. Get into the proper form! ˆ ŷ =or For emple, if ou re given ŷ 9 6 = 44, ˆ = divide oth sides 44 first nd fctor into squres: = 9 44 6 44 = 44/9 If ou re given something like 44/6 = 6 8 = 9 =. 4 3 seprte the stu nd the stu nd complete squres, nd del with shifting coordintes if necessr.
9 Find the vertices, foci, nd smptotes of the hperol nd sketch its grph. 9.. 44 5 6 36. 4. 9 4 36 3. 3 4 8 4. 6 9 64 9 35 5 3 Identif the tpe of conic section whose eqution is given nd find the vertices nd foci. 5. 6. 7. 4 8. 8 6 6 9. 4 3 3. 4 4
6 REVIEW OF CONIC SECTIONS EXERCISES 8 Find the verte, focus, nd directri of the prol nd sketch its grph... 4 3. 4 4. 5. 8 3 6. 5 7. 8. 9 Find n eqution of the prol. Then find the focus nd directri. 9.. 6 Find the vertices nd foci of the ellipse nd sketch its grph... 9 5 3. 4 6 4. 4 5 5 5. 9 8 4 7 6. A 5 6 Click here for nswers. _ 6 4 7 64 7 8 Find n eqution of the ellipse. Then find its foci. S Click here for solutions. 3. 4. 3 4 8 6 9 64 9 35 5 3 Identif the tpe of conic section whose eqution is given nd find the vertices nd foci. 5. 6. 7. 4 8. 8 6 6 9. 4 3 3. 4 4 3 48 Find n eqution for the conic tht stisfies the given conditions. 3. Prol, verte,, focus, 3. Prol, verte,, directri 5 33. Prol, focus 4,, directri 34. Prol, focus 3, 6, verte 3, 35. Prol, verte,, is the -is, pssing through (, 4) 36. Prol, verticl is, pssing through, 3,, 3, nd, 9 37. Ellipse, foci,, vertices 5, 38. Ellipse, foci, 5, vertices, 3 39. Ellipse, foci,,, 6 vertices,,, 8 4. Ellipse, foci,, 8,, verte 9, 4. Ellipse, center,, focus,, verte 5, 4. Ellipse, foci,, pssing through, 43. Hperol, foci, 3, vertices, 44. Hperol, foci 6,, vertices 4, 7. 8. 45. Hperol, foci, 3 nd 7, 3, vertices, 3 nd 6, 3 46. Hperol, foci, nd, 8, vertices, nd, 6 47. Hperol, vertices 3,, smptotes 48. Hperol, foci, nd 6,, smptotes nd 6 Thomson Brooks-Cole copright 7 9 Find the vertices, foci, nd smptotes of the hperol nd sketch its grph. 9.. 44 5 6 36. 4. 9 4 36 49. The point in lunr orit nerest the surfce of the moon is clled perilune nd the point frthest from the surfce is clled polune. The Apollo spcecrft ws plced in n ellipticl lunr orit with perilune ltitude km nd polune ltitude 34 km (ove the moon). Find n eqution of this ellipse if the rdius of the moon is 78 km nd the center of the moon is t one focus.