- Parametric Equations 9. A hperbola with foci (, ) and (6, ) and vertices (, ) and (, ).. An ellipse with foci (, ) and (, 6) and vertices (, ) and (, ).. A parabola with ais the ais and passing through the points (, ) and (, ).. A parabola with verte at (6, ), ais the line, and passing through the point (, 7).. An ellipse with vertices (, ), and (, ) that passes through the origin.. A hperbola with vertices at (, ), and (, ) that passes through the point (, ). C In Problems, find the coordinates of an foci relative to the original coordinate sstem:. Problem 6. Problem 6 7. Problem 7. Problem 9. Problem. Problem In Problems, use a graphing utilit to find the coordinates of all points of intersection to two decimal places.. 7, 6. 7 9,. 7,., 6 SECTION - Parametric Equations Parametric Equations and Plane Curves Projectile Motion Ccloid Parametric Equations and Plane Curves Consider the two equations t t t 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. TABLE t FIGURE Graph of t, t t, t. 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
Additional Topics in Analtic Geometr Then, substituting the result into the second equation, we obtain ( ) ( ) We recognize this as the equation of a parabola, as we would guess from Figure. 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 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 t t t t t () 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.
- Parametric Equations EXAMPLE Graphing Parametric Equations and Eliminating the Parameter Graph the plane curve given parametricall b cos sin () Identif the curve b eliminating the parameter. Solution Construct a table and graph: /6 / / / /6 7/6 / / / /6 FIGURE Graph of cos, sin,. To eliminate the parameter, we solve the first equation in () for cos, the second for sin, and substitute into the Pthagorean identit cos sin : The graph is an ellipse (Fig. ). cos and sin cos sin 6 6 Matched Problem Graph the plane curve given parametricall b cos, sin,. Identif the curve b eliminating the parameter. EXPLORE-DISCUSS Graph one period ( ) of each of the three plane curves given parametricall b cos sin cos sin cos sin Identif the curves b eliminating the parameter.
Additional Topics in Analtic Geometr EXAMPLE Parametric Equations for Conic Sections Find parametric equations for the conic section with the given equation: (A) 9 (B) 6 7 Solutions (A) B completing the square in and we obtain the standard form ( ) ( ). So the graph is an ellipse with center (, ) and 9 major ais on the line. Since cos sin, a parametric representation with parameter is obtained b letting cos, sin : cos sin (B) B completing the square in and we obtain the standard form ( ) ( ). So the graph is a hperbola with center (, ) and 6 transverse ais on the line. Since sec tan, a parametric representation with parameter is obtained b letting sec, tan : sec tan, k, k an integer FIGURE sec, tan. Note that when the parametric equations are graphed using a graphing utilit in connected mode, the graph appears to show the asmptotes of the hperbola (see Fig. ). Matched Problem Find parametric equations for the conic section with the given equation: (A) 6 6 96 (B) 6 9 6 76 Projectile Motion Newton s laws and advanced mathematics can be used to determine the path of a projectile. If v is the initial speed of the projectile at an angle with the horizontal (see Fig. ) and air resistance is neglected, then the path of the projectile is given b (v cos )t (v sin )t.9t t b ()
- Parametric Equations The parameter t represents time in seconds, and and are distances measured in meters. Solving the first equation in () for t in terms of, substituting into the second equation, and simplifing results in the following equation:.9 (tan ) (6) v cos You should verif this b suppling the omitted details. FIGURE Projectile motion. v v cos v sin We recognize equation (6) 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 () 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. The range of a projectile is the distance from the point of firing to the point of impact. If we keep the initial speed v of the projectile constant and var the angle in Figure, 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,. 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. ). FIGURE 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 Figure 6).
6 Additional Topics in Analtic Geometr FIGURE 6 Ccloid. P(, ) a C Q O R S Since the circle rolls along the ais without slipping (refer to Figure 6), we see that d(o, S) arc PS a in radians (7) where S is the point of contact between the circle and the ais. Referring to triangle CPQ, we see that Using these results, we have d(o, R) d(o, S) d(r, S) (arc PS) d(p, Q) d(p, Q) a sin / () d(q, C) a cos / (9) a a sin Use equations (7) and (). d(r, P) d(s, C) d(q, C) a a cos Use equation (9) and the fact that d(s, C) a. Even though in equations () and (9) was restricted so that /, it can be shown that the derived parametric equations generate the whole ccloid for. The graph specifies a periodic function with period a. Thus, in general, we have Theorem. Theorem Parametric Equations for a Ccloid 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
- Parametric Equations 7 P Q FIGURE 7 Ccloid path. 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. 7). EXPLORE-DISCUSS (A) Let Q be a point b units from the center of a wheel of radius a, where b a. If the wheel rolls along the ais with the tracing point Q starting at (, 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. 6; circle. (A) 7 cos, 6 sin, (B) tan, sec,, k, k an integer EXERCISE - A In Problems, plot each plane curve b use of a table of values (see Eample ). Obtain an equation in and b eliminating the parameter, and identif the curve. (In this eercise set, the interval for the parameter is the whole real line, unless stated to the contrar.). t, t. t, t. t, t. t, t. t, t 6. t, t 7. t, t. t, t 9. t, t. t, t B In Problems, obtain an equation in and b eliminating the parameter. Use the simpler of the two forms to plot the curve. Name the curve if it is a curve we have identified.. sin, cos. sin, cos. sin, cos. sin, cos. t, ; t t
Additional Topics in Analtic Geometr 6. t, ; t t 7. t, t; t. t, t 9. If A, C, and E, find parametric equations for A C D E F. Identif the curve.. If A, C, and D, find parametric equations for A C D E F. Identif the curve. C In Problems 6, obtain an equation in and b eliminating the parameter. Use the simpler of the two forms to plot the curve. Name the curve if it is a curve we have identified.. t, t ; t. e t, e t. cos, sin. sec, tan. 6. t t, t, t t t t Graph, using a calculator, one period ( ) of each ccloid in Problems 7 and. 7. sin, cos. sin, cos In Problems 9, use a graphing utilit to graph the parametric equations. Then eliminate the parameter and find the standard equation for the curve. Name the curve and find its center. 9. 6 cos t, sin t, t. sec t, tan t, t, t. tan t, sec t, t, t. cos t, sin t, t. Find an equation of the form A C D E F that has the same graph as the parametric equations tan t, tan t,. t. Repeat Problem for cot t, (t cot t)/t, t, t. In Problems, find the standard form for each equation. Name the curve and find its center. Use parametric equations to graph the curve on a graphing utilit.. 9 66 6. 6 6 76 7. 9 9 6. 6 9 6 6 APPLICATIONS 9. Plane Motion. An object follows a path as given b sin 6t cos 6t where t is time in seconds and and are distances in feet. (A) What are the coordinates of the object when t. second? Compute answers to one decimal place. (B) Eliminate the parameter and graph the resulting equation in and. Identif the path.. Plane Motion. Repeat Problem 9 for sin t cos t t t. Projectile Motion. A projectile is fired with an initial speed of 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 using a calculator.. Projectile Motion. Repeat Problem if the same projectile is fired at to the horizontal instead of. CHAPTER GROUP ACTIVITY Focal Chords Man of the applications of the conic sections are based on their reflective or focal properties. One of the interesting algebraic properties of the conic sections concerns their focal chords. If a line through a focus F contains two points G and H of a conic section, then the line segment GH is called a focal chord. Let G(, ) and H(, ) be points on the graph of a such that GH is a focal chord. Let u denote the length of GF and v the length of FH (see Fig. ).
Chapter Review 9 FIGURE Focal chord GH of the parabola a. G F u v H (a, a) (A) Use the distance formula to show that u a. (B) Show that G and H lie on the line a m, where m ( )/( ). (C) Solve a m for and substitute in a, obtaining a quadratic equation in. Eplain wh a. (D) Show that. u v a (u a) (E) Show that u v a. Eplain wh this implies that u v a, with equalit if and onl u a if u v a. (F) Which focal chord is the shortest? Is there a longest focal chord? (G) Is a constant for focal chords of the ellipse? For focal chords of the hperbola? Obtain evidence u v for our answers b considering specific eamples. (H) The conic section with focus at the origin, directri the line D, and eccentricit E has the DE polar equation r. Eplain how this polar equation makes it eas to show that u v E cos a for a parabola. Use the polar equation to determine the sum for a focal chord of an ellipse or u v hperbola. Chapter Review - CONIC SECTIONS; PARABOLA The plane curves obtained b intersecting a right circular cone with a plane are called conic sections. If the plane cuts clear through one nappe, then the intersection curve is called a circle if the plane is perpendicular to the ais and an ellipse if the plane is not perpendicular to the ais. If a plane cuts onl one nappe, but does not cut clear through, then the intersection curve is called a parabola. If a plane cuts through both nappes, but not through the verte, the resulting intersection curve is called a hperbola. A plane passing through the verte of the cone produces a degenerate conic a point, a line, or a pair of lines. The figure illustrates the four nondegenerate conics. Circle Ellipse Parabola Hperbola