88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative to the original coordinate sstem. 7. Problem 8. Problem 6 9. Problem 7 0. Problem 8. Problem 2 2. Problem 22 (2, 0) In Problems 6, find the coordinates of all points of intersection to two decimal places. (, ) (, ) (2, 2). 2 2 7 2 0, 6. 8 2 2 7 9 0, 2. 7 2 8 2 0, 2 2 2 0 6. 2 2 2 2 0, 2 2 6 2 0 Section 0- Parametric Equations Parametric Equations and Plane Curves Parametric Equations and Conic Sections Projectile Motion Ccloid FIGURE Graph of t, t 2 2t, t. 0 Parametric Equations and Plane Curves Consider the two equations t t 2 2t t () Each value of t determines a value of, a value of, and hence, an ordered pair (, ). To graph the set of ordered pairs (, ) determined b letting t assume all real values, we construct Table listing selected values of t and the corresponding values of and. Then we plot the ordered pairs (, ) and connect them with a continuous curve, as shown in Figure. The variable t is called a parameter and does not appear on the graph. Equations () are called parametric equations because both and are epressed in terms of the parameter t. The graph of the ordered pairs (, ) is called a plane curve.
0- Parametric Equations 89 T A B L E t 0 2 2 2 0 0 0 8 8 Parametric equations can also be graphed on a graphing utilit. Figure 2(a) shows the Parametric mode selected on a Teas Instruments TI-8 calculator. Figure 2(b) shows the equation editor with the parametric equations in () entered as T and T. In Figure 2(c), notice that there are three new window variables, Tmin, Tma, and Tstep, that must be entered b the user. FIGURE 2 Graphing parametric equations on a graphing utilit. 0 7 2 (a) (b) (c) (d) Eplore/Discuss (A) Consult the manual for our graphing utilit and reproduce Figure 2(a). (B) Discuss the effect of using different values for Tmin and Tma. Tr Tmin and. Tr Tma and. (C) Discuss the effect of using different values for Tstep. Tr Tstep, 0., and 0.0. In some cases it is possible to eliminate the parameter b solving one of the equations for t and substituting into the other. In the eample just considered, solving the first equation for t in terms of, we have t Then, substituting the result into the second equation, we obtain ( ) 2 2( ) 2 We recognize this as the equation of a parabola, as we would guess from Figure.
820 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY In other cases, it ma not be eas or possible to eliminate the parameter to obtain an equation in just and. For eample, for t log t t e t t 0 ou will not find it possible to solve either equation for t in terms of functions we have considered. Is there more than one parametric representation for a plane curve? The answer is es. In fact, there is an unlimited number of parametric representations for the same plane curve. The following are two additional representations of the parabola in Figure. t t 2 2t t t 2 t t (2) t () The concepts introduced in the preceding discussion are summarized in Definition. DEFINITION PARAMETRIC EQUATIONS AND PLANE CURVES A plane curve is the set of points (, ) determined b the parametric equations f(t) g(t) where the parameter t varies over an interval I and the functions f and g are both defined on the interval I. Wh are we interested in parametric representations of plane curves? It turns out that this approach is more general than using equations with two variables as we have been doing. In addition, the approach generalizes to curves in three- and higher-dimensional spaces. Other important reasons for using parametric representations of plane curves will be brought out in the discussion and eamples that follow. EXAMPLE Eliminating the Parameter Eliminate the parameter and identif the plane curve given parametricall b t 9 t 0 t 9 ()
0- Parametric Equations 82 Solution To eliminate the parameter t, we solve each equation in () for t: t 2 t 9 t 2 9 t t 9 2 Equating the last two equations, we have 2 9 2 2 2 9 A circle of radius centered at (0, 0) Thus, the graph of the parametric equations in () is the quarter of the circle of radius centered at the origin that lies in the first quadrant (Fig. ). FIGURE.. (a) (b) MATCHED PROBLEM Eliminate the parameter and identif the plane curve given parametricall b t, t, 0 t. Parametric Equations and Conic Sections Trigonometric functions provide ver effective representations for man conic sections. The following eamples illustrate the basic concepts. EXAMPLE 2 Identifing a Conic Section in Parametric Form Eliminate the parameter and identif the plane curve given b 8 cos sin 0 2 () Solution To eliminate the parameter, we solve the first equation in () for cos, the second for sin, and substitute into the Pthagorean identit cos 2 sin 2 : cos and sin 8
822 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY FIGURE Graph of 8 cos, sin, 0 2. 9 6 9 cos 2 sin 2 8 2 2 2 2 6 6 The graph is an ellipse (Fig. ). 6 MATCHED PROBLEM 2 Eliminate the parameter and identif the plane curve given b cos, sin, 0 2. Eplore/Discuss 2 Graph one period (0 2) of each of the three plane curves given parametricall b cos 2 2 cos cos sin 2 2 sin 2 sin Identif the curves b eliminating the parameter. What happens if ou graph less than one period? More than one period? EXAMPLE Solutions FIGURE 2 cos, sin, 0 2. 9 9 9 Parametric Equations for Conic Sections Find parametric equations for the conic section with the given equation: (A) 2 2 9 2 00 0 (B) 2 6 2 0 2 7 0 (A) B completing the square in and we obtain the standard form ( 2) 2 ( )2. So the graph is an ellipse with center (2, ) and 9 2 major ais on the line 2. Since cos 2 sin 2, a parametric 2 representation with parameter is obtained b letting cos, sin : 2 cos sin Since sin and cos have period 2, graphing these equations for 0 2 will produce a complete graph of the ellipse (Fig. ).
0- Parametric Equations 82 FIGURE 6 sec, tan, 0 2,. 2, 2 6 2. 2. (B) B completing the square in and we obtain the standard form ( ) 2 ( ) 2. So the graph is a hperbola with center (, ) and 6 transverse ais on the line. Since sec 2 tan 2, a parametric representation with parameter is obtained b letting sec, tan : sec tan The period of tan is, but the period of sec is 2, so we have to use 0 2 to produce a complete graph of the hperbola (Fig. 6). To be precise, we should eclude /2 and /2, since the tangent function is not defined at these values. Including them does not affect the graph, since most graphing utilities ignore points where functions are undefined. Note that when the parametric equations are graphed in the connected mode, the graph appears to show the asmptotes of the hperbola (see Fig. 6). MATCHED PROBLEM Find parametric equations for the conic section with the given equation. (A) 6 2 6 2 0 96,2 0 (B) 6 2 9 2 6 28 76 0 Remark Refer to Eample, part A. An interval of the form a a b, where b 2, will produce a graph containing all the points on this ellipse. We will follow the practice of alwas choosing the shortest interval starting at 0 that will generate all the points on a conic section. For this ellipse, that interval is [0, 2]. Projectile Motion Newton s laws and advanced mathematics can be used to determine the path of a projectile. If v 0 is the initial speed of the projectile at an angle with the horizontal and a 0 is the initial altitude of the projectile (see Fig. 7), then, neglecting air resistance, the path of the projectile is given b (v 0 cos )t 0 t b (6) a 0 (v 0 sin )t.9t 2 The parameter t represents time in seconds, and and are distances measured in meters. Solving the first equation in equations (6) for t in terms of, substituting into the second equation, and simplifing, produces the following equation:.9 a 0 (tan ) 2 (7) v 2 0 cos 2 You should verif this b suppling the omitted details.
82 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY FIGURE 7 Projectile motion. v 0 a 0 v 0 cos v 0 sin We recognize equation (7) as a parabola. This equation in and describes the path the projectile follows but tells us little else about its flight. On the other hand, the parametric equations (6) not onl determine the path of the projectile but also tell us where it is at an time t. Furthermore, using concepts from phsics and calculus, the parametric equations can be used to determine the velocit and acceleration of the projectile at an time t. This illustrates another advantage of using parametric representations of plane curves. EXAMPLE Projectile Motion An automobile drives off a 0-meter cliff traveling at 2 meters per second (see Fig. 8). When (to the nearest tenth of a second) will the automobile strike the ground? How far (to the nearest meter) from the base of the cliff is the point of impact? FIGURE 8 0 m Solution At the instant the automobile leaves the cliff, the velocit is 2 meters per second, the angle with the horizontal is 0, and the altitude is 0 meters. Substituting these values in equations (6), the parametric equations for the path of the automobile are 2t 0.9t 2 The automobile strikes the ground when 0. Using the parametric equation for, we have 0.9t 2 0.9t 2 0 0 t.2 seconds.9
0- Parametric Equations 82 The distance from the base of the cliff is the same as the value of. Substituting t.2 in the first parametric equation, the distance from the base of the cliff at the point of impact is 2(.2) 80 meters. MATCHED PROBLEM A gardener is holding a hose in a horizontal position. meters above the ground. Water is leaving the hose at a speed of meters per second. What is the distance (to the nearest tenth of a meter) from the gardener s feet to the point where the water hits the ground? The range of a projectile at an altitude a 0 0 is the distance from the point of firing to the point of impact. If we keep the initial speed v 0 of the projectile constant and var the angle in Figure 7, we obtain different parabolic paths followed b the projectile and different ranges. The maimum range is obtained when. Furthermore, assuming that the projectile alwas stas in the same vertical plane, then there are points in the air and on the ground that the projectile cannot reach, irrespective of the angle used, 0 80. Using more advanced mathematics, it can be shown that the reachable region is separated from the nonreachable region b a parabola called an envelope of the other parabolas (see Fig. 9). FIGURE 9 Reachable region of a projectile. Envelope Ccloid We now consider an unusual curve called a ccloid, which has a fairl simple parametric representation and a ver complicated representation in terms of and onl. The path traced b a point on the rim of a circle that rolls along a line is called a ccloid. To derive parametric equations for a ccloid we roll a circle of radius a along the ais with the tracing point P on the rim starting at the origin (see Fig. 0). FIGURE 0 Ccloid. P(, ) a C Q O R S
826 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Since the circle rolls along the ais without slipping (refer to Fig. 0), we see that d(o, S) arc PS a in radians (8) where S is the point of contact between the circle and the ais. Referring to triangle CPQ, we see that d(p, Q) a sin 0 /2 (9) d(q, C) a cos 0 /2 (0) Using these results, we have d(o, R) d(o, S) d(r, S) (arc PS) d(p, Q) a a sin Use equations (8) and (9). d(r, P) d(s, C) d(q, C) a a cos Use equation (0) and the fact that d(s, C ) a. Even though in equations (9) and (0) was restricted so that 0 /2, it can be shown that the derived parametric equations generate the whole ccloid for. The graph specifies a periodic function with period 2a. Thus, in general, we have Theorem. THEOREM PARAMETRIC EQUATIONS FOR A CYCLOID For a circle of radius a rolled along the ais, the resulting ccloid generated b a point on the rim starting at the origin is given b a a sin a a cos P FIGURE Ccloid path. Q The ccloid is a good eample of a curve that is ver difficult to represent without the use of a parameter. A ccloid has a ver interesting phsical propert. An object sliding without friction from a point P to a point Q lower than P, but not on the same vertical line as P, will arrive at Q in a shorter time traveling along a ccloid than on an other path (see Fig. ).
0- Parametric Equations 827 Eplore/Discuss (A) Let Q be a point b units from the center of a wheel of radius a, where 0 b a. If the wheel rolls along the ais with the tracing point Q starting at (0, a b), eplain wh parametric equations for the path of Q are given b a b sin a b cos (B) Use a graphing utilit to graph the paths of a point on the rim of a wheel of radius, and a point halfwa between the rim and center, as the wheel makes two complete revolutions rolling along the ais. Answers to Matched Problems. The quarter of the circle of radius 2 centered at the origin that lies in the fourth quadrant. 2. 2 2 6, circle of radius centered at (0, 0). (A) Ellipse: 7 cos, 6 sin, 0 2 (B) Hperbola: 2 tan, sec, 0 2, 2, 2. 2.8 meters EXERCISE 0- A. If t 2 and t 2 2, then 2. Discuss the differences between the graph of the parametric equations and the graph of the line 2. 2. If t 2 and t 2, then 2 2. Discuss the differences between the graph of the parametric equations and the graph of the parabola 2 2. In Problems 2, the interval for the parameter is the whole real line. For each pair of parametric equations, eliminate the parameter t and find an equation for the curve in terms of and. Identif and graph the curve.. t, 2t 2. t, t. t 2, 2t 2 2 6. t 2, t 2 7. t, 2t 8. 2t, t 9. t 2, t 0. 2t, t 2. t, t 2 2. 2t 2, t B In Problems 2, obtain an equation in and b eliminating the parameter. Identif the curve.. t 2, 2t. t, 2t 2. t, t, t 0 6. t, t, t 0 7. t, 26 t, 0 t 6 8. t, 2 t, 0 t 2 9. t, t, t 20. 2 t, t, t 2 2. sin, cos, 0 2 22. sin, cos, 0 2 2. 2 2 sin, 2 cos, 0 2 2. sin, 2 2 cos, 0 2 2. If A 0, C 0, and E 0, find parametric equations for A 2 C 2 D E F 0. Identif the curve.
828 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 26. If A 0, C 0, and D 0, find parametric equations for A 2 C 2 D E F 0. Identif the curve. In Problems 27 0, eliminate the parameter and find the standard equation for the curve. Name the curve and find its center. 27. 6 cos t, 2 sin t, 0 t 2 28. sec t, 2 2 tan t, 0 t 2, t 29. 2 tan t, sec t, 0 t 2, t 2, 2 0. cos t, 8 sin t, 0 t 2 C In Problems 6, the interval for the parameter is the entire real line. Obtain an equation in and b eliminating the parameter and identif the curve.. t 2, t 2 9 2. t 2, t 2 2 2t., t 2 t 2 t., t 2 t 2 8 t., t 2 t 2 t t 2 6., t 2 t 2 In Problems 7 0, find the standard form of each equation. Name the curve and find its center. Then use trigonometric functions to find parametric equations for the curve. 7. 2 2 200 9 2 8 66 0 8. 6 2 60 2 8 760 0 9. 2 2 9 2 92 62 0 0. 6 2 2 9 2 6 6 0 2, 2. Consider the following two pairs of parametric equations:. t, e t, t 2. 2 e t, 2 t, t (A) Graph both pairs of parametric equations in a squared viewing window and discuss the relationship between the graphs. (B) Eliminate the parameter and epress each equation as a function of. How are these functions related? 2. Consider the following two pairs of parametric equations:. t, log t, t 0 2. 2 log t, 2 t, t 0 (A) Graph both pairs of parametric equations in a squared viewing window and discuss the relationship between the graphs. (B) Eliminate the parameter and epress each equation as a function of. How are these functions related? APPLICATIONS. Projectile Motion. An airplane fling at an altitude of,000 meters is dropping medical supplies to hurricane victims on an island. The path of the plane is horizontal, the speed is 2 meters per second, and the supplies are dropped at the instant the plane crosses the shoreline. How far inland (to the nearest meter) will the supplies land?. Projectile Motion. One stone is dropped verticall from the top of a tower 0 meters high. A second stone is thrown horizontall from the top of the tower with a speed of 0 meters per second. How far apart (to the nearest tenth of a meter) are the stones when the land?. Projectile Motion. A projectile is fired with an initial speed of 00 meters per second at an angle of to the horizontal. Neglecting air resistance, find (A) The time of impact (B) The horizontal distance covered (range) in meters and kilometers at time of impact (C) The maimum height in meters of the projectile Compute all answers to three decimal places. 6. Projectile Motion. Repeat Problem if the same projectile is fired at 0 to the horizontal instead of.