Section.8 Grphs of Polr Equtions 98 9. Points:,,,,.,... The points re colliner. 9. Points:.,,.,,.,... not colliner. Section.8 Grphs of Polr Equtions When grphing polr equtions:. Test for symmetry. () ) or (r, ). Polr xis: Rlce (r, ) by (r, ) or (r, ). Pole: Rlce (r, ) by (r, ) or (r, ). (d) r f (sin ) is symmetric with respect to the line (e) r f (cos ) is symmetric with respect to the polr xis.. Find the vlues for which r is mximum.. Find the vlues for which r.. Know the different types of polr grphs. () Limçons <, < b Rose curves, n Circles (d) Lemnisctes r ± b cos r cos n r cos r cos r ± b sin r sin n r sin r sin : Rlce (r, ) by (r,. Plot dditionl points.. r Vocbulry Check.. polr xis. convex limçon. circle. lemniscte. crdioid. r cos Rose curve with petls. r sin Crdioid. r cos Limçon with inner loop. r cos Lemniscte. r sin Rose curve with petls. r cos Circle 7. r cos Polr xis: Pole: : Answer: r cos() r cos Not n equivlent eqution r cos() r cos Equivlent eqution r cos Not n equivlent eqution polr xis 8. r cos r cos : r cos r cos Not n equivlent eqution Polr r cos xis: r cos r cos Equivlent eqution Pole: r cos Not n equivlent eqution Answer: polr xis
98 Chpter Topics in Anlytic Geometry 9. r sin r : sin( ) r sin cos cos sin r sin Equivlent eqution Polr xis: r sin() r sin Not n equivlent eqution Pole: r sin Answer:. r cos Polr xis: Pole: r : Not n equivlent eqution r cos cos Equivlent eqution r cos Not n equivlent eqution Answer: polr xis. r cos r cos : r cos Equivlent eqution Polr xis: r cos Pole: r cos Equivlent eqution r cos r cos Equivlent eqution Answer: the, polr xis, nd the pole. r sin Polr xis: Pole: r sin : Not n equivlent eqution r sin Not n equivlent eqution r sin Equivlent eqution Answer: pole. r ( sin ) sin sin sin sin Mximum: r () or sin sin Not possible. r cos cos cos 8 cos Mximum: r 8 cos ( sin ) cos sin Zero: r, Zero: r,
Section.8 Grphs of Polr Equtions 987. cos r cos cos cos ±,,. r sin sin sin sin ±,,, 7 Mximum: r,, Mximum: r,,, 7 cos sin cos,, Zero: r,, sin,,, Zero: r,,, 7. Circle: r 8. Circle: r 9. Circle: r. r. r sin Circle with rdius of. r cos polr xis Circle with rdius. r cos the polr xis b Crdioid r r. r sin b r 8 Crdioid r
988 Chpter Topics in Anlytic Geometry. r sin 7.. r cos b Crdioid r 8 r polr xis b Crdioid r r r sin Limçon with b < inner loop r 9 r 7, 8 8. r sin, b b r 7 Dimpled limçon 9. r sin Limçon with b < inner loop r r,. r cos the polr xis b r r Limçon with inner loop,. r cos the polr xis Limçon with inner loop b < r 7 r cos or.7,.
Section.8 Grphs of Polr Equtions 989. r cos the polr xis b > Dimpled limçon r 7. r sin nd the pole Rose curve n with petls r r,,,,,, 7 the polr xis, 8. r cos the polr xis Rose curve n with four petls r,,, r,,, 7. r sec r cos r cos x Line. r csc r sin y Line 7. r r(sin cos ) y x sin cos y x Line 8. r r sin cos y x sin cos y x Line
99 Chpter Topics in Anlytic Geometry 9. r 9 cos the polr xis, nd the pole Lemniscte,. r sin r sin r sin. r 8 cos min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl =. r cos min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl =. r ( sin ). min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl = r cos min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl =. r 8 sin cos min = mx = st = Xmin = - Xmx = Xscl = Ymin = - Ymx = Yscl =. r csc sin 9 9 8 min = mx = st = Xmin = -9 Xmx = 9 Xscl = Ymin = - Ymx = 8 Yscl = 7. r cos < 8. r cos < 7 9. r cos < 7 7
Section.8 Grphs of Polr Equtions 99. r sin. < r 9 sin <. r < <..... r sec cos r cos cos r(r cos ) r cos r ±x y x x ±x y ±x y (x ) x ±x y x x x y x (x ) y x x (x ) x x (x ) x x x x (x ) (x ) x x x x x x (x ) (x ) y ± x x x (x ) ± x x x x The grph hs n symptote t x.. ±x y y y r csc sin r sin sin rr sin r sin r ±x y y y ±x y ±x y x y y y y y x y y y y x ± y y y y The grph hs n symptote t y. ± y y y y
99 Chpter Topics in Anlytic Geometry. r r sin sin r sin y. r cos sec r cos sin cos cos cos y sin As, y r cos cos sin x cos sin As, x. 7. True. For grph to hve polr xis symmetry, rlce r, by r, or r,. 8. Flse. For grph to be symmetric bout the pole, one portion of the grph coincides with the other portion rotted rdins bout the pole. 9. r cos () Upper hlf of circle 7 Lower hlf of circle 7 (d) Entire circle Left hlf of circle 7 7. r cos () 8 8 9 8 8 The ngle hs the effect of rotting the grph by the ngle. For prt, r cos sin.. Let the curve r f() be rotted by to form the curve r g(). If r, is point on r f(), then r, is on r g(). Tht is, g r f. Letting, or, we see tht g g f f. (, r θ + φ) φ (, r θ) θ
Section.8 Grphs of Polr Equtions 99. Use the result of Exercise. () Rottion: Originl grph: r fsin Rotted grph: r f sin Rottion: Originl grph: r fsin Rotted grph: r f sin fsin fcos Rottion: Originl grph: r fsin Rotted grph: r f sin fcos. () r sin sin cos cos sin (sin cos ) r sin( ) sin cos cos sin sin (d) r sin sin cos cos sin cos r sin sin cos cos sin cos. r sin () r sin sin cos sin cos r sin sin cos sin cos (d) r sin sin sin sin cos sin cos r sin sin sin sin cos sin cos. () r sin r sin Rotte the grph in prt () through the ngle.
99 Chpter Topics in Anlytic Geometry. () r sec r cos r cos x r cos r sin r sec r r cos cos cos sin sin x y (d) r cos r sin r sec r r x y r r cos r sec cos cos sin sin cos cos sin sin cos r sin y 7. r k sin 7 8. r sin k k : k : r Circle r sin Convex limçon 7 k = k = k = k = 8 () r sin. < r sin. < k : r sin k : Crdioid r sin Limçon with inner loop Yes. r sink. Find the minimum vlue of, >, tht is multiple of tht mkes multiple of. k
Section.9 Polr Equtions of Conics 99 9. y x 9 x x 9 x x 9 x 9 x ± 7. y x No zeros 7. x x x x y x 7. y x 7 x Zero: x x x 7. Vertices:,,, Center t, nd 7. Foci:,,, ; Mjor xis of length 8 Minor xis of length : Horizontl mjor xis b b y Center: h, k, Verticl mjor xis x h x 9 y k b y x, c, b c 9 b 7 x h b x 7 y k y y 7 x Section.9 Polr Equtions of Conics The grph of polr eqution of the form r is conic, where e > is the eccentricity nd is the distnce between the focus (pole) nd the directrix. () If e <, the grph is n ellipse. If e, the grph is prbol. If e >, the grph is hyperbol. Guidelines for finding polr equtions of conics: () (d) or r ± e cos ± e sin Horizontl directrix bove the pole: r e sin Horizontl directrix below the pole: r e sin Verticl directrix to the right of the pole: r e cos p Verticl directrix to the left of the pole: r e cos