# 12th - CHAPTER 10 CONIC SECTIONS AND POLAR COORDINATES 10.1...

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Unformatted text preview: CHAPTER 10 CONIC SECTIONS AND POLAR COORDINATES 10.1 CONIC SECTIONS AND QUADRATIC EQUATIONS 4. 312‘; :> 413:2 :> 1):. :focusis(0_., l». I.— ) ,dircctrix is y : — l». I.— [C |—‘ ”' “L‘s? Y—f—E —> 49-8 p—2; focus is [0, —2}, dircclrix is y = 2 23. 6x3—9y3254 2 xii—£21 :;> c: V303 — b3 : \f9— f3: 32. 39—3)? :3 :> —x3: I :> c: VaB—b2 : «v.33 — I : 2; asymptotcs are 3! : ; VG): 42. (a) — = 1 :5» center is (0, 0), vertices are (0,5) and (0, —5);c = V 33 — I.)2 = V/l— = 4 2:» foeiare (0, 4) and (0, —4) ; therefore the new center is (—3, —2), the new vertices a1'e(—3_. 3) and (—3, —7), and the new foci are (—3, 2) and (—3, —6) x3 ,2— 50. 7—. — (— l_. 0) and (1, 0); therefore the new center is (3,4), the new vertices are (3 ; V5.4 (it-313 , 2_ ., —(3 —4) —l are (2, 4) and (4,4); the new equation is 71. 10.2 CLASSIl-"YING CONIC SECTIONS BY I‘ICCENTRICITY l 2:» center is (0.0), vertices are (V/Eﬁ) and (—V/Eﬁ) ;c = \, /32_b2: x’_—1:]::> foeiare 9. Foei:[0,;3],e 0.5 »c 3anda E 6 »b3 36—9 27 20. UsingPF:e-PD,wehave V(x—4)3—y3= |x—9| 2:» (x—4)2—y2= zg‘LxB—mx—sn=>ng—ygzzo25x3—9y32130m§—%=1. 1 s \r an the new foei 23 y3—3x~=3:.~ “Ti—x32] :czv’aB—tﬁ =\f3—l=2:>e=§=%;asymptotesare \} y = _ V/gx, F[0_. ; 2] ;direetriees are y = 0;: — 2 _3 (vs) 37.Focus(—2,0)andDirectrixx —.]3 »c ae 2andg 13 »e2 »e PF—ZPD V50: 2)? (y (DB—2x 33 (x :3 y3—4(x 33]? x3 4): y? 24(x3 x 3.):2 ye: 32:»):2 :12] 10.3 QUA DRA TIC. EQUATIONS AND ROTATIONS 12. 232 — 4.9X)’ — 3y2 — 4x = 7 2:» B2 — 4AC = (—4.9)2 — 4(2)(3) = 0.0] '9 CI 2:» Hyperbola 15. 6x3 — 3xy — 2y? —17y— 2 = 0 2 B2 — 4AC = 32 — 4mm) 2 —39 < 0 :» Ellipse _ q_ . I 22. cot 2a- % — 713 » 2n- 27” » a %;tliereforex= x’ cos a— 5” sin _ x _ . ﬁ ﬁ y:x’slna—}Hcosa::>x=.X’—l%}" y=l2—'x’—% I) X’ ‘f)”)(‘-.3x’ éy’) ifjd=l=>4rﬂ=l I_ ma => 3(%x’—H—’?y)2—2 G(. l 2:» Parallel horizontal lines 10.4 CONICS AND PARAMETRIC EQUATIONS; THIC CYCLUID 7. x=—sect,y=tant,—%<t<% I) 2:» secgt—tangtzl 2:» xg—y‘zl 12. x: sinht,y=2cosht,—oo<t<oo ﬂ 2:» 4COSl12l—4Slllh3l24 3333—31224 3' 32—: -4 4. (a) (3,? — Zmr] and (—3, 57” — Enrr] , n an integer y (b) (—3, g — anr] and (3, 57" — EnTr] , n an integer (c) (3, — i — 2n7r] and (—3, 3?” — 211a] , n an integer (d) (—3, — g — inn] and (3, 37” — 211a] ,n an integer I Hum ta.—ml o o 6. (a) x : VG cos § : l, y : p6 sin § : l 2:» Cartesian coordinates are (l, l) (b) x — 1 cos 0 — l, y — 1 sin 0 — 0 -*» Cartesian coordinates are (l, 0) (c) x — 0 cos — 0, y — 0 sin — 0 -*» Cartesian coordinates are (0.0) (d) x = —\,/§cos = — l, y = —\,/3 sin 2 —1 2:» Cartesian coordinates are (— l, — l) (e) x = 3 cos 5‘; = 3g? , y = 3 sin 5‘; = 2:» Cartesian coordinates are (355, — (f) x = 5 cos (Ian—1 31] = 3, y = 5 sin (tan—l = 4 2:» Cartesian coordinates are (3.4) (g) x = —1 cos Trr : l, y = —1 sin "in = 0 2:» Cartesian coordinates are (1,0) (ll) x = 2V 3 cos = —\,/3, y = 2V 3 sin = 3 ::> Cartesian coordinates are (—V/g, 3) 13. E 3 31. 13 : 1 :> x2 — y? : 1, Circle with center C : (0,0) and radius 1 Six—19:3 2:» roosﬁ—rsinﬁz'j 10.6 GRAPHING IN POLAR COORDINATES 10. sinﬁr — 9) = sin F) = 13 2:» (1".TI' — 9) and (—r,?r — 9) are on the graph when (139) is on the graph ::> symmetric about the y—axis and the x—axis; therefore symmetric about the origin 19.62%31'213(1,%:];6‘:—%:>r:_] 2’(“>—%J;9=%w=—I244%); 62—3?”::,1.:]:> ]!_T,r]; J_\$_ o. 1—da—2005_6‘, Slope = r’siné‘ircosé‘ _ 2005235in3ircos3 r'cosﬂ—rsinﬂ _ Zoosﬁcosg—rsiné' JI' 2:» Slope at (l, . Zoos[— sin[— “:1 H—iJcos[— Slope at( 1’ l5 2005L— cos L—E)—E—]JsinL—;) 31r isNOSE]sin(%jn—ncos(‘wj _ I J J J 1—1 510 eat I. T p ( ' 4 5 -(—l]sin(37"' :1; 4 Slope at (I, 3”] is 2 cos (— sin (— tilicos E— l 24. (a) (b) 26. 1‘ 2 sec 0 r b) rcosl’} 2 x cos 5‘ U 35. (x5) :4sinl’) => szsino => u:%,%zmints of intersection an: _. and _. . The points (VG, — and (VG, — alt found by graphing. ...
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## This note was uploaded on 05/19/2008 for the course MS 4032 taught by Professor Anony during the Spring '08 term at A.T. Still University.

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12th - CHAPTER 10 CONIC SECTIONS AND POLAR COORDINATES 10.1...

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