CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.................... Section. Plane Curves and Parametric Equations.......... Section. Parametric Equations and Calculus............ 9 Section. Polar Coordinates and Polar Graphs............ Section. Area and Arc Length in Polar Coordinates........ Section. Polar Equations of Conics and Kepler s Laws....... Review Eercises............................. Problem Solving..............................
CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.. Verte:, p > Opens to the right Matches graph (h).. Verte:, p < Opens downward Matches graph (e). 9 Center:, Ellipse Matches (f) 7. Hperbola Center:, Vertical transverse ais Matches (c) 9.. Verte:, 8 Focus:, Directri: (, ) 8 Verte: Focus:,., Directri:.7 (, ) 8 8.. Verte:, Focus:, Directri: (, ) Verte:, Focus:, Directri: (, ) 7. 9., Verte: Focus:, Directri: Verte:, Focus:, Directri: ±. 8
Chapter Conics, Parametric Equations, and Polar Coordinates. h p k. 9 7. Since the ais of the parabola is vertical, the form of the equation is a b c. Now, substituting the values of the given coordinates into this equation, we obtain c, 9a b c, a b c. Solving this sstem, we have a, Therefore, b, c. or 9. 9. a, b, c (, ) Center:, Foci: ±, Vertices: ±, e. 9 a, b 9, c Center:, Foci:, 9,, Vertices:,,, 8 (, ) 8. 9 9 9 a 9, b, c 9 e Center:, Foci:, ± Vertices:,,, (, ) e. 7 7 a, b, c Center:, Foci: ±, Vertices: ±, Solve for : 7 7 7 ± (Graph each of these separatel.)
Section. Conics and Calculus 7.. 9 9 a, b, c Center:, Foci: ±, Vertices:,, 7, Solve for : 7 7 ± 8 (Graph each of these separatel.) 9. Center:, Focus:, Verte:, Horizontal major ais a, c b 9. Vertices:,,, 9 Minor ais length: Vertical major ais Center:, a, b 9. Center: Horizontal major ais Points on ellipse:,,, Since the major ais is horizontal,,. a b. Substituting the values of the coordinates of the given points into this equation, we have a, b, c Center:, Vertices:, ± Foci:, ± Asmptotes: ± 9 a b, and. a The solution to this sstem is a, b 7. Therefore, 7,. 7 7. a, b, c Center:, Vertices:,,, Foci: ±, Asmptotes: ±
Chapter Conics, Parametric Equations, and Polar Coordinates 9. 9 8. 9 8 9 9 8 9 9 9 8 8 a, b, c 9 9 ± Center:, Vertices:,,, Foci: ±, Asmptotes: ± Degenerate hperbola is two lines intersecting at,.. 9 9 9 8 8 8 a, b, c Center:, Vertices:, ± Foci:, ± Solve for : 9 9 8 9 9 7 ± 9 7 (Graph each curve separatel.). 7 9 7 8 a, b, c Center:, Vertices:,,, Foci: ±, 7. Vertices: ±, Asmptotes: ± Horizontal transverse ais Center:, a, ± b a ±b ± b Therefore,. 9 Solve for : 9 7 8 9 ± 7 7 (Graph each curve separatel.)
Section. Conics and Calculus 7 9. Vertices:, ± Point on graph:, Vertical transverse ais Center:, a. Center: Verte: Focus:,,, Vertical transverse ais a, c, b c a Therefore,. Therefore, the equation is of the form. 9 b Substituting the coordinates of the point,, we have 9 b or b 9. Therefore, the equation is. 9 9. Vertices: Asmptotes: Horizontal transverse ais Center:, a,,, Slopes of asmptotes:, ± b a ± Thus, b. Therefore, 9.. (a) 9, 9, From part (a) we know that the slopes of the normal 9 lines must be 9. ± At : ±, 9 9 At, : 9 At, : or 9 or At, : ± 9 9 or At, : 9 or 9 7. 7. 9. 9 9 Ellipse 7. Circle (Ellipse) Parabola 9 9 9 9 9 Circle (Ellipse) 9 9 7. ) Hperbola 77. (a) A parabola is the set of all points, that are equidistant from a fied line (directri) and a fied point (focus) not on the line. h p k or k p h (c) See Theorem..
8 Chapter Conics, Parametric Equations, and Polar Coordinates 79. (a) A hperbola is the set of all points, for which the absolute value of the difference between the distances from two distance fied points (foci) is constant. (c) h a k ± b h a k b or k a or k ± a h b h b 8. Assume that the verte is at the origin. The pipe is located 9 p p 9 p meters from the verte. (, ) (, ) Focus 8. a a The equation of the tangent line is a a or a a. Let. Then: a a a a a Therefore, = a, is the -intercept. (, a) (, ) 8. (a) Consider the parabola p. Let m be the slope of the one tangent line at, and therefore, m is the slope of the second at,. Differentiating, p or and we have: p, m p or pm m p or p. m Substituting these values of into the equation p, we have the coordinates of the points of tangenc pm, pm and pm, pm and the equations of the tangent lines are pm m pm and The point of intersection of these lines is pm, p m ( ) p p, m m p p m m m. = p and is on the directri, p. 8 Verte: At,, At d. Tangent line at, : Tangent line at Since, d Point of intersection: d, :. m m,,, d.. the lines are perpendicular. = p ( pm, pm ) ( ) pm ( ), p m Directri: and the point of intersection lies on this line.,
Section. Conics and Calculus 9 87. d At the point of tangenc, on the mountain, m. Also, Choosing the positive value for, we have. Thus, m m ±. The closest the receiver can be to the hill is.. ± ± m. (, ) (, ) (, ) 89. Parabola Verte:, p p p 8 8 Circle Center:, k Radius: 8 k k k 8 k (Center is on the negative -ais.) ± Since the -value is positive when, we have. A arcsin 8 8 arcsin d. square feet
Chapter Conics, Parametric Equations, and Polar Coordinates 9. (a) Assume that a. a a 8 8 f 8, S f 9 9 d 9 9 9 ln 9 9,7 9 ln,7 9 ln 9 9 8 9 ln 9 ln 9 9 ln 9 9 9 d 9 ln 8. m (Formula ) (, ) (, ) 9. p, p,,,, As p increases, the graph becomes wider. p = p = p = 9. (a) At the vertices we notice that the string is horizontal and has a length of a. The thumbtacks are located at the foci and the length of string is the constant sum of the distances from the foci. 97. 7 8 9 7 9 8 7 7 p = p = 8 8 Focus Focus Verte Verte 99. e c a A P a a A P c a P A P P A P A c P ( a, ) e c a A P A P A P A P. e A P.9.9.97 A P.9.9
Section. Conics and Calculus.. 9 9 8 9 8 At 8, : The equation of the tangent line is 8. It will cross the -ais when and 8. when. At, or. Endpoints of major ais: At, or. Endpoints of minor ais: Note: Equation of ellipse is 8 9 is undefined when. 9 9 8,,,,,, 7. (a) A d Disk: arcsin V d 8 or, A ab S d (c) Shell: arcsin 9 9.8 S 8 ln V d d ln d.9 8
Chapter Conics, Parametric Equations, and Polar Coordinates 9. From Eample, For C a e sin d, 9 we have a 7, b, c 9, e c a 7. C 7 9 sin d 8.8 7.9. Area circle r Area ellipse ab a a a Hence, the length of the major ais is a.. The transverse ais is horizontal since, and, are the foci (see definition of hperbola). Center:, c, a, b c a 7 Therefore, the equation is 9 7.. a a c b 7 8 9 7 9 8 7 7 7. Time for sound of bullet hitting target to reach c Time for sound of rifle to reach, : Since the times are the same, we have:, : c v m c v s v s c v m c v s c v s ( c, ) rifle (, ) (, c ) target c c c v m v m v c c s v s v s c v m v s c v s v m v m v s v s v m c c v s v m c v m v s v m 9. The point, lies on the line between, and,. Thus,. The point also lies on the hperbola. Using substitution, we have: 9 7 7 8 7 8 ± 8 77 7 8 ± 9 9 ± 9 7 Choosing the positive value for we have: 9 9 7 9 7.8 and.
Section. Conics and Calculus. Let c a b. a b b a. b a c a a b b b a b There are four points of intersection: At a b b a a a b a b ± aa b ac ± a b a b a b b c b h b c ac b the slopes of the tangent lines are: a b, a b, e a b a a b b a b ± b a b a b b a b e b a b ac b a c a a b a b a b ac a b, ± b a b, and h b ac b c a c. a b a b ac a b, ± b a b Since the slopes are negative reciprocals, the tangent lines are perpendicular. Similarl, the curves are perpendicular at the other three points of intersection.. False. See the definition of a parabola.. True 7. True 9. Let be the equation of the ellipse with a > b >. Let ±c, be the foci, a b c a b. Let u, v be a point on the tangent line at P,, as indicated in the figure. Now, b a a b b a b a v u b a a a v b b u a b a v b u a b a v b u Slope at P, c a b F F c b d (u, v) P(, ) a a + b = CONTINUED
Chapter Conics, Parametric Equations, and Polar Coordinates 9. CONTINUED Since there is a right angle at u, v, We have two equations: Multipling the first b v and the second b u, and adding, Similarl, v u a b vb a u. a v b u a b a u b v. a v a u a b v u v b v d b v u d a. v d b. From the figure, u d cos and v d sin. Thus, cos d and cos sin d a d b b d a d a b a sin d b. Let r PF and r PF, d a b b a r r a. r r r r r r a c c a c a b a Finall, d r r b a b a b a b b b a a a b a b b a b a a a b b a b a b a b a b a b a b, a constant!
Section. Plane Curves and Parametric Equations Section. Plane Curves and Parametric Equations. t, t (a) t (c) (d) t,. t. t 7. t t t t t implies t 9. t, t. t. t, t t t t 8 8. e t, > e t, >
Chapter Conics, Parametric Equations, and Polar Coordinates 7. sec 9. cos,. sin sin cos <, < Squaring both equations and adding, we have 9. cos sin cos,. cos. cos 7. sec sin sin tan cos cos sec sin sin 9 tan 9 8 9 9 9. t. e t ln t e t ln ln e t e t > >
Section. Plane Curves and Parametric Equations 7. B eliminating the parameters in (a) (d), we get. The differ from each other in orientation and in restricted domains. These curves are all smooth ecept for. (a) t, t cos cos d when d d, ±, ±,.... (c) e t e t (d) e t e t > > > > <.. The curves are identical on < The are both smooth. Represent 7. (a) The orientation of the second curve is reversed. (c) The orientation will be reversed. (d) Man answers possible. For eample, t, t, and t, t. 9. t t t m. h a h a k b cos sin k b h a cos k b sin. From Eercise 9 we have t t. Solution not unique. From Eercise we have cos sin. Solution not unique 7. From Eercise we have a, c b cos sin. Center:, Solution not unique
8 Chapter Conics, Parametric Equations, and Polar Coordinates 9. From Eercise we have.. a, c b Eample Eample sec t, t t, t tan. t, t t, t Center:, Solution not unique tan t, tan t. sin 9. 7. sin cos cos cos sin Not smooth at n 7 Not smooth at, ±, and, ±, or n.. cot. See definition on page 79. sin Smooth everwhere. A plane curve C, represented b f t, gt, is smooth if and are continuous and not simultaneousl. See page 7. f g 7. When the circle has rolled radians, we know that the center is at a, a. sin sin8 C b BD b cos cos8 AP or AP b cos b Therefore, a b sin and a b cos. or BD b sin P b A C θ a B D 9. False t t The graph of the parametric equations is onl a portion of the line.
Section. Parametric Equations and Calculus 9 7. (a) (c) mihr 8 h v sin t t It is not a home run when, <. v cos t cos t sin t t ftsec (d) We need to find the angle (and time t) such that From the first equation t cos. Substituting into the second equation, We now solve the quadratic for tan : tan.8 cos t sin t t. sin cos cos 7 tan sec tan tan. tan tan 7 9. Yes, it s a home run when, >. Section. Parametric Equations and Calculus. dt d ddt t t. d d dd cos sin sin cos Note: and d. t, t 7. t, t t dt d ddt t when t. d d d Line d d concave upwards 9. cos, sin. sec, tan d cos cot when sin. d sec sec tan d d csc csc sin when. sec tan csc when. concave downward d d d d d d csc cot sec tan cot when. concave downward
Chapter Conics, Parametric Equations, and Polar Coordinates. cos, sin. cot, sin d sin cos cos sin d sin cos sin csc cos tan when. d d sec cos sin cos sin At,, Tangent line:, and d 8. 8 sec csc when concave upward. At,, 8 8, and. d Tangent line: At,,, and d 8. Tangent line: 8 8 7. t, t, t 9. t t, t t, t (a) (a) 8 At t,,,, (d) d, dt, dt. d (, ) and (c). At,, d. At t,,,, (d) d dt, (, ), dt. d and (c). At,, d 8. sin t, sin t crosses itself at the origin,,,. At this point, t or t. cos t d cos t At t : and Tangent Line d. At t, and Tangent Line d.
Section. Parametric Equations and Calculus. t t, t t crosses itself at the point,,. At this point, t or t. d t t At t, and. Tangent Line d At t, and or. Tangent Line dt 9. cos Horizontal tangents: Vertical tangents: Note: d sin, when Points:, n,, n where n is an integer. Points shown:,, when corresponds to the cusp at,,. sin cos d d sin d sin cos cos tan at ± ±,, ± ±,, ±,... ±,... Points: n n, Points shown:,,,,, n,,, 7. t, t 9. t, t t Horizontal tangents: t when t dt Horizontal tangents: when t ±. dt t Point:, Points:,,, Vertical tangents: d ; dt none Vertical tangents: d ; dt none (, ) (, ) (, ). cos, sin Horizontal tangents: cos when Points: Vertical tangents: Points: d,,, d sin when d,,,,.,.. cos, sin Horizontal tangents: cos when Points: Vertical tangents: Points: d,,,,,, d sin when d,.,. (, ) (, ) 8 (, ) (, ) (, ) (, ) (, ) (, )
Chapter Conics, Parametric Equations, and Polar Coordinates. sec, tan Horizontal tangents: sec ; none d (, ) (, ) Vertical tangents: d sec tan when,. d Points:,,, 7. t, t t 9. d t t tt t d d t t t 8t t t Concave upward for t > Concave downward for t < t ln t, t ln t, t > t t d t t d d t t t t t t t t t d Because t >, d > Concave upward for t >. sin t, cos t, < t <. d sin t cos t tan t d t d sec cos t cos t Concave upward on < t < Concave downward on < t < t t, t, t d t, t dt dt b s d a dt dt dt t t dt 8t t 9t dt t t dt. e t, t, t 7. t, t, t d dt et, dt b s d a dt dt dt e t dt d t, dt, dt s t dt d dt dt t t tt lnt t ln.9
Section. Parametric Equations and Calculus 9. e t cos t, e t sin t, t d dt et cos t sin t dt et sin t cos t, s d dt dt dt e t dt e t e. e t dt. t, s u du ln u u u u ln7 7.9 u t, t, t 9 dt du t dt d dt t, dt t dt t. d a cos,. a sin a cos sin a sin cos d s 9a cos sin 9a sin cos d a sin cos cos sin d a d sin d a cos a a sin, d a cos, a sin d s a cos a sin a cos d sin a cos d a cos a cos, d 8a d 7. 9 cos t, (a) Range: 9. ft, t 9 sin t t (c) d 9 cos, 9 sin t dt dt for s 9 cos 9 sin t dt.8 ft t. 9. t t t, t (a) (c) s t t t t dt t 8 t 8 t t t t t dt.7 dt t 8t t t t t t t Points:,, when t or t.,.799,. t t 8 t t t t dt
Chapter Conics, Parametric Equations, and Polar Coordinates. (a) t sin t cos t t t sint cost t The average speed of the particle on the second path is twice the average speed of a particle on the first path. (c) t t sin cos t The time required for the particle to traverse the same path is t.. d t, dt. t, dt S t dt.9 7t dt 87 d cos, cos sin cos, sin S cos cos sin sin d. d d 7. d t, t,, dt dt 9. (a) S t dt t dt t S t dt t t dt d cos, sin, sin, cos S cos sin cos d d d cos d sin 7. d a cos, a sin, a cos sin, a sin cos S d d a sin 9a cos sin 9a sin cos d a sin a cos d sin a dt 7. d ddt See Theorem.7. 7. One possible answer is the graph given b t, t. 77. s b d a dt dt dt See Theorem.8.
Section. Parametric Equations and Calculus 79. Let be a continuous function of on a b. Suppose that f t, gt, and ft a, Then using integration b substitution, d ft dt and b d t gtft dt. a t f t b. 8. sin sin tan d sin cos d A sin tan sin cos d 8 sin cos 8 sin d 8 sin cos 8 θ < 8. ab8. is area of ellipse (d). a is area of cardioid (f). 87. ab is area of hourglass (a). 8 89. 9. t, A t, d A A d,, 8 < t < d t t dt tt t t dt t d cos, sin, sin d V sin sin d sin d cos sin d cos cos t t dt 8t tt dt (Sphere) t dt t t t 8t t dt t tt t t 8 9. a sin, a cos (a) d a sin, a cos d d d a sin a cos sin cos d d cos cos sin sin cos a cos cos a cos acos CONTINUED At, d. Tangent line: a a, a, a
Chapter Conics, Parametric Equations, and Polar Coordinates 9. CONTINUED (c) sin, cos d sin cos Points of horizontal tangenc:, an, a (d) Concave downward on all open -intervals:...,,,,,,,... (e) a sin a cos d a cos d s sin d a sin d a a cos 8a 9. t u r cos r sin rcos sin v w r sin r cos rsin cos r w θ r θ t v u θ (, ) 97. (a) t t, t t, The graph for < t < is the circle, ecept the point,. Verif: t t t t t t t t t t (c) As t increases from t to, the speed increases, and as t increases from to, the speed decreases. d 99. False. d dt gt d ft ft ftgt gtft ft Section. Polar Coordinates and Polar Graphs.,. cos sin,, ( ),, cos sin,, ( ),
Section. Polar Coordinates and Polar Graphs 7.,. 7. r,, 9. cos..,.,. sin..99,.,.99 (.,.) r,.,.,.8,.9 (,.) (.8,.9).,,.,,.,, r ± r ±9 ± r tan tan tan,,,,,.,.,,.,,. r,,, (, ) (, ) (, ) 7.,, 9. r,.,.88,, r,.8,.9. (a),,. r,,. (,.) (,.). r sin circle Matches (c). r cos Cardioid Matches (a) 7. a 9. r a r sin r csc a
8 Chapter Conics, Parametric Equations, and Polar Coordinates.. 9 r cos r sin r sin 9r cos r cos sin r cos sin r 9 cos sin r 9 csc cos 7. r 7. r sin 9. r 9 r r sin 9 r tan r tan tan arctan. r sec. r cos. r sin r cos < < 7. r cos Traced out once on < < 9. r cos <
Section. Polar Coordinates and Polar Graphs 9. r sin r sin r sin <. r h cos k sin r rh cos k sin r hr cos kr sin h k h k h h k k h k h k h k Radius: Center: h, k h k.,,, 7. d cos cos.,., 7,. d 7 7 cos.. 8 cos.7. 9. d cos sin cos sin cos cos sin sin At, At,, At r sin cos sin cos sin cos sin cos sin,,,. d d.. d. (a), 8 r cos r,,,, Tangent line: (c) At, d... (a), r sin (c) At,.7. d r,,, Tangent line:, 9 9 9
Chapter Conics, Parametric Equations, and Polar Coordinates. r sin 7. r csc sin cos cos sin d cos sin cos or sin Horizontal tangents:,,,,, d sin sin cos cos d,,, csc cos csc cot sin d cos, Horizontal:,,, sin sin sin sin sin sin sin sin or sin, 7, Vertical tangents:, 7,, 9. r sin cos 7. r csc Horizontal tangents: r,,,.,.78,.,. Horizontal tangents: r, 7,,, 7. r sin r r sin 7. r sin Cardioid Smmetric to -ais, 9 Circle r Center:, Tangent at the pole: 77. r cos Rose curve with three petals Smmetric to the polar ais Relative etrema:,,,, r, Tangents at the pole:,,
Section. Polar Coordinates and Polar Graphs 79. r sin Rose curve with four petals Smmetric to the polar ais, Relative etrema: ±, Tangents at the pole: give the same tangents.,,, ±, and pole, 8. r 8. r cos Circle radius: Cardioid 8. r cos Limaçon 87. r csc r sin Smmetric to polar ais r Horizontal line 89. r 9. r cos Spiral of Archimedes Smmetric to Tangent at the pole: r r cos, Lemniscate Smmetric to the polar ais, Relative etrema: ±, Tangents at the pole: r ± ±, and pole,
Chapter Conics, Parametric Equations, and Polar Coordinates 9. Since the graph has polar ais smmetr and the tangents at the pole are,. Furthermore, r as Also, r sec, cos r as r r r.. r r r cos r cos Thus, r ± as. = 9. r Hperbolic spiral r as lim r r sin sin r sin sin sin lim = cos 97. The rectangular coordinate sstem consists of all points of the form, where is the directed distance from the -ais to the point, and is the directed distance from the -ais to the point. Ever point has a unique representation. The polar coordinate sstem uses r, to designate the location of a point. r is the directed distance to the origin and measured clockwise. Points do not have a unique polar representation. is the angle the point makes with the positive -ais, 99. r a, circle b, line. r sin (a) (c). Let the curve r f be rotated b to form the curve r g. If r is a point on r f, then r,, is on r g. That is, g r f. (, r θ + φ), Letting, or we see that g g f f. φ (, r θ ) θ
Section. Polar Coordinates and Polar Graphs. r sin (a) r sin sin cos r sin cos cos (c) r sin sin sin (d) r sin cos 7. (a) r sin r sin Rotate the graph of r sin through the angle. 9. tan r At, drd cos sin tan is undefined.. r cos tan r At ψ drd, tan cot. arctan θ cos cot sin 8.. r tan r cos cos dr sin sin cos At dr d cos cos sin cos d tan,,. ψ θ. True 7. True
Chapter Conics, Parametric Equations, and Polar Coordinates Section. Area and Arc Length in Polar Coordinates. A f d sin d sin d. A f d sin d. (a) r 8 sin A 8 sin d sin d cos d sin A A 7. cos d sin 9. A cos d. sin 8 A sin d cos sin. A cos d sin sin. The area inside the outer loop is cos d sin sin From the result of Eercise, the area between the loops is A..
Section. Area and Arc Length in Polar Coordinates 7. r cos 9. r cos r cos r sin Solving simultaneousl, Solving simultaneousl, cos cos cos sin cos cos sin,. Replacing r b r and b in the first equation and solving, cos cos, cos, Both curves pass through the pole,,, and,, respectivel. Points of intersection:,,,,,. tan, 7. Replacing r b r and b in the first equation and solving, cos sin, sin cos, which has no solution. Both curves pass through the pole,,, and,, respectivel. Points of intersection:,,, 7,,. r sin r sin Solving simultaneousl, sin sin. r r Solving simultaneousl, we have sin,.,. Points of intersection:,,, Both curves pass through the pole,, arcsin, and,, respectivel. Points of intersection:,,,,,. r sin r r sin is the equation of a rose curve with four petals and is smmetric to the polar ais, and the pole. Also, r is the equation of a circle of radius centered at the pole. Solving simultaneousl,, sin,,. Therefore, the points of intersection for one petal are, and,. B smmetr, the other points of intersection are, 7,,,,,, 7,, 9, and,.
Chapter Conics, Parametric Equations, and Polar Coordinates 7. r cos r sec The graph of r cos is a limaçon with an inner loop b > a and is smmetric to the polar ais. The graph of r sec is the vertical line. Therefore, there are four points of intersection. Solving simultaneousl, cos sec cos cos cos ± arccos arccos.7.8. Points of intersection:.8, ±.7,.8, ±.7 r = sec θ 8 r =+ cosθ 9. r cos r sin Points of intersection:,,.9,.,.,. The graphs reach the pole at different times ( values). r = cos θ r = sin θ. From Eercise, the points of intersection for one petal are, and,. The area within one petal is sin d r = d sin d. 8 sin d sin. Total area A sin d d cos sin r = + sin θ (b smmetr of the petal). A sin 8 r = sin θ sin d d 9 9 r = sin θ r = sin θ r =
7. A a a a cos d sin a a sin a a Section. Area and Arc Length in Polar Coordinates 7 9. a A 8 a cos d a a 8 cos cos d a 8 a 8 a a sin sin a a a. (a) r a cos r ar cos a a = a = (c) A cos cos d cos d cos d cos cos d sin 8 sin. r a cosn For n : For n : For n : r a cos A a a r a cos A 8 a cos d a r a cos A a cos d a a a a For n : r a cos A 8 a cos d a a In general, the area of the region enclosed b r a cosn for n,,,... is a if n is odd and is a if n is even.
8 Chapter Conics, Parametric Equations, and Polar Coordinates. r a 7. r s a d a (circumference of circle of radius a) a r sin r cos s sin cos d sin d cos sin sin 8 d 9. r,. r,.. r sin cos,.. Length.. Length.7 Length.9. r cos 7. r e a r sin S cos sin cos sin d 7 sin cos d sin r ae a S e a cos e a ae a d a e a cos d a ea a a cos sin a a ea a 9. r cos r 8 sin S cos sin cos sin d cos sin cos sin d.87. Area f d r d Arc length f f d r dr d d. (a) is correct: s..
Section. Area and Arc Length in Polar Coordinates 9. Revolve r about the line r sec. f, f S cos d cos d sin r = r = sec cos 7. r 8 cos, (a) A r d (Area circle r ) cos d cos....8... A.. 7..8.7..8 d sin (c), (d) For of area For of area For of area.7:. 8.:.7 7.7:.7 (e) No, it does not depend on the radius. 9. r a sin b cos r ar sin br cos a b b a represents a circle. 7. (a) r, As a increases, the spiral opens more rapidl. If <, the spiral is reflected about the -ais. r a,, crosses the polar ais for n and integer. To see this for r a r sin a sin n. n, The points are r, an, n, n,,,.... (c) (d) f, f s ln. A r dr d ln d
Chapter Conics, Parametric Equations, and Polar Coordinates 7. The smaller circle has equation r a cos. The area of the shaded lune is: a A a a a cos d cos d sin r = = r = acos This equals the area of the square, a a a a a Smaller circle: r cos. 7. False. f and g have the same graphs. 77. In parametric form, b s d a dt dt dt. Using instead of t, we have r cos f cos and r sin f sin. Thus, d f cos f sin and f sin f cos. d It follows that d d d f f. Therefore, s d f f d. Section. Polar Equations of Conics and Kepler s Laws. r (a) (c) e e cos e, r e., r e., r, cos parabola,. cos cos,. cos cos ellipse hperbola e =. 8 e =. e =.. r (a) (c) e e sin e, r, parabola 9 9 sin e., r e., r,. sin sin,. sin sin ellipse hperbola e =. e =. e =. 8
Section. Polar Equations of Conics and Kepler s Laws. r e sin (a) e =. e =. e =. e =.7 e =.9 The conic is an ellipse. As e, the ellipse becomes more elliptical, and as e, it becomes more circular. (c) 9 8 e =. e =. e = 9 The conic is a hperbola. As e, the hperbolas opens more slowl, and as e, the open more rapidl. The conic is a parabola. 7. Parabola; Matches (c) 9. Hperbola; Matches (a). Ellipse; Matches. r sin. r cos cos Parabola because e ; d Distance from pole to directri: d Directri: Verte: r,, Ellipse because e ; d Directri: Distance from pole to directri: d Vertices: r,,,, 7. r sin 9. r sin sin Ellipse because e ; d Directri: Distance from pole to directri: Vertices: r,,,, d r cos cos Hperbola because e > ; d Directri: Distance from pole to directri: Vertices: d r,,,, 8. r sin sin Hperbola because e > ; d Directri: d Distance from pole to directri: Vertices: r, 8,,,
Chapter Conics, Parametric Equations, and Polar Coordinates. r sin. r cos Ellipse Parabola 7. r sin Rotate the graph of r sin 9. r cos Rotate the graph of r cos. Change to : r cos counterclockwise through the angle. clockwise through the angle. 8. Parabola e,, d r ed e cos cos. Ellipse e,, d r ed e sin sin sin 7. Hperbola e,, d r ed e cos cos 9. Parabola Verte:, e, d, r sin. Ellipse Vertices:,, 8, e, d r ed e cos cos cos. Hperbola Vertices: e, d 9 r,, 9, ed e sin 9 sin 9 sin. Ellipse if < e <, parabola if e, hperbola if e >.
Section. Polar Equations of Conics and Kepler s Laws 7. (a) Hperbola e > Ellipse e < (c) Parabola e (d) Rotated hperbola e 9. r b cos a cos a b r a b b a a b b r cos a r sin a b r a cos b a a b a b a b a cos a b a c cos. a, c, e, b. r 9 cos a, b, c, e r 9 cos b ca cos b e cos. A 9 cos d cos d.88 7. Vertices: a c,9. 9,. e c a r : r (,, 7,, 9, 9, 7, 9,88.977,9 ed e cos r ed e, a,9. 7979.,,,.977 cos,9,88 cos,9. : r ed e ed ed e e,9 d e e e d,9 e e e d e e 8.97 When r,., Distance between earth and the satellite is r, miles. 9. a.9 8, e.7. r e a 9,8,78. e cos.7 cos Perihelion distance: a e 7,,8 km Aphelion distance: a e,98, km a.9 9, e.88 r e a,,,9 e cos.88 cos Perihelion distance: a e,,87, km Aphelion distance: a e 7,7,,8 km
Chapter Conics, Parametric Equations, and Polar Coordinates. r (a). 9.88 cos A 9 r d 9.8 8 km 8 9 r d 9.8 8 B trial and error, r d.89 rs r d.98.98 > 9.9 because the ras in part (a) are longer than those in part. (c) For part (a) 9 s r drd d. 9 km Average per ear For part.98 s Average per ear. 9.89 r drd d. 9. 9.89.7 8 kmr.89 8 kmr. r a c, r a c, r r c, r r a e c a r r r r c e e a c a c a c r r a ed ed 7. r and r sin sin Points of intersection: ed,, ed, ed d sin cos ed cos sin sin r : ed sin sin ed cos sin cos At ed,, At d. ed d sin cos ed cos sin sin r : ed sin sin ed cos sin cos At ed,,. At ed,, d d. Therefore, at ed, we have m m, and at ed, we have m m. The curves intersect at right angles. ed,,. d Review Eercises for Chapter.. Ellipse Verte:,. Matches (e). Parabola opening to left. Matches Ellipse Verte:, Matches (a)
Review Eercises for Chapter 7. 9. 9 9 Circle Center: Radius:,, 8 9 8 8 Hperbola Center:, Vertices: ±, Asmptotes: ±. 9 9 9 8 Ellipse Center:, Vertices:, ± (, ). Verte:, Directri: Parabola opens to the right. p. Vertices:,, 7, Foci:,,, Horizontal major ais Center:, a, c, b 7. Vertices: ±, 9. Foci: ±, Center:, Horizontal transverse ais a, c, b, a, b, c, e 9 B Eample of Section., C 9 sin d.87.. has a slope of. The perpendicular slope is. when and d. Perpendicular line: 7
Chapter Conics, Parametric Equations, and Polar Coordinates. (a) Focus:,, S d 8,9.9,. t, t t Line 7. cos, sin 9. sec, tan Circle sec tan Hperbola 8 8. t t. t t (other answers possible) Let 9 cos and 9 sin. Then cos and sin.. cos cos sin sin 7 8 7. t t (a) d No horizontal tangents (c) t
Review Eercises for Chapter 7 9. t t (a) d t t (c) No horizontal tangents, t t. t t t (a) d t t t t Point of horizontal tangenc: t t t t when t. t t,. cos sin cos (a). cot when, d sin. Points of horizontal tangenc:, 7,, (c) 8 8, (c). cos sin (a) d sin cos cos sin sin tan cos (c) when,. d But, at Hence no points of dt dt horizontal tangenc.,.
8 Chapter Conics, Parametric Equations, and Polar Coordinates 7. t, t 9. d dt, t dt for t. dt Horizontal tangent at t :,, No vertical tangents sin, cos d cos, sin d for d Horizontal tangents:,,,, d for,,... Vertical tangents:,,,, d d,,,.... cot sin sin cos. rcos rsin sin cos (a), (c) d r cos d At, d,, d d and d. r sin s r cos sin d r r d d r. t, t, t 7. d, dt (a) S t dt t S dt, d dt dt 9 dt t sin, cos A b a sin d cos cos cos d d 9. r,,, cos, sin,. ( ), r,,., cos., sin..87,.79 (,.).,, r 7 r,, 7,, (, )
Review Eercises for Chapter 9. r cos 7. r cos r r cos r r cos ± 9. r cos cos sin r r cos r sin 7. r cos sec 7. a 7. a arctan cos cos r cos 8 cos r ar cos r sin r a cos sin r a 8 77. r 79. r sec Circle of radius r cos, Centered at the pole Smmetric to polar ais, Vertical line cos, and pole 8. r cos 8. r cos Cardioid Limaçon Smmetric to polar ais Smmetric to polar ais r r 7 8. r cos Rose curve with four petals Smmetric to polar ais, Relative etrema: Tangents at the pole:,,, and pole,,,,,,
Chapter Conics, Parametric Equations, and Polar Coordinates 87. r sin r ± sin Rose curve with four petals Smmetric to the polar ais, and pole, Relative etrema: Tangents at the pole: ±,,, ±, 89. r cos Graph of r sec rotated through an angle of 8 9. r cos sec Strophoid Smmetric to the polar ais r as r as 9. r cos (a) The graph has polar smmetr and the tangents at the pole are (c) d sin cos cos sin cos cos sin Horizontal tangents: cos ± cos, cos 8 When, r 8 8,, arccos 8.8,.8, arccos 8.8,.8, arccos.8, 8., arccos 8.8,.. cos ± Vertical tangents:. sin cos, sin, cos,, ±arccos., ±.8,. ± 8,, ±arccos,,,,.
Review Eercises for Chapter 9. Circle: r sin d cos sin sin cos cos cos sin sin Limaçon: r sin at d cos sin sin cos cos cos sin sin Let be the angle between the curves: tan 9. Therefore, arctan 9.. sin cos sin tan at, d 9, d 97. r cos, r cos The points, and, are the two points of intersection (other than the pole). The slope of the graph of r cos is m d r sin r cos sin cos cos r cos r sin At,, m and at,, m. The slope of the graph of r cos is m d sin cos cos sin cos sin cos. sin cos sin cos. At,, m and at,, m. In both cases, m m and we conclude that the graphs are orthogonal at, and,. 99. r cos A cos d., 9. r sin A sin d. r sin cos A., sin cos d..... r, r 8 sin 9 r 8 sin sin A 8 sin d.8 9.8.8.8 9 d 8 sin d
Chapter Conics, Parametric Equations, and Polar Coordinates 7. r a cos, dr a sin d s a cos a sin d a cos d sin a cos d a cos a 9. f cos f sin f S f cos sin 7 8 cos cos sin 7 8 cos d 7 88.8. r, e. sin Parabola r Ellipse e cos cos, 8. r e sin sin, Hperbola 7. Circle Center:,, in rectangular coordinates Solution point:, r r sin r sin 9. Parabola Verte: Focus: e, d r,, cos. Ellipse Vertices:,,, Focus:, a, c, e, d r cos cos
Problem Solving for Chapter Problem Solving for Chapter. (a) (c) Intersection: 8 (, ) (, ) 8, Point of intersection,,, is on directri. Tangent line at, Tangent line at Tangent lines have slopes of and perpendicular.,. Consider p with focus F, p. Let Pa, b be point on parabola. B A p p F P(a, b) b a a p Tangent line at P Q For, b a a pb a b b p p p b. Thus, Q, b. FQP is isosceles because FQ p b FP a b p a b bp p pb b bp p b p b p. Thus, FQP BPA FPQ.. (a) In OCB, cos a OB OB a sec. In OAC, cos OA a OA a cos. r OP AB OB OA asec cos a cos cos a sin cos a tan sin CONTINUED
Chapter Conics, Parametric Equations, and Polar Coordinates. CONTINUED r cos a tan sin cos a sin (c) r a tan sin r sin a tan sin sin a tan sin, < Let t tan, < t <. Then sin t t t and a t, a t t. < r cos a sin r cos a r sin a a + t t θ 7. (a) t t t, t t r sin r cos r cos r cos t t t t t t sin r cos cos r cos r cos sin sin cos r cos r cos sin cos cos sin Thus,. r cos cos r cos sec (c) (d) r for,. Thus, and are tangent lines to curve at the origin. (e) 9. (a) t t t t t t t t t t t t t ±, ± ± ± ± t, u cos t is on du, u sin du the curve whenever, is on the curve. (c) t t cos, t t sin, t t Generated b Mathematica Thus, s a On,, s. dt a.
Problem Solving for Chapter. r ab, a sin b cos ra sin b cos ab a b ab b a. Let r, be on the graph. r r cos r r cos r r cos r r r cos r r cos Line segment r cos Area ab r cos r cos. (a) The first plane makes an angle of 7 with the positive -ais, and is miles from P: cos 7 7t sin 7 7t Similarl for the second plane, cos 9 t cos 9 t sin 9 t sin 9 t. d (c) 8 The minimum distance is 7.9 miles when t.. cos 9 t cos 7 7t sin 9 t sin 7 7t 7. n = n = n = n = n = n = n = n = n = n = n = n,,,, produce bells ; n,,,, produce hearts.