744 CHAPTER 10 Analytic Geometry Either equations (3) or equations (4) can serve as parametric equations for the ellipse x y 9 1. 2 2 + = The direction of the motion, the beginning point, and the time for 1 revolution give a particular parametric representation. Figure 75 x y 9 1 2 2 + = with counterclockwise orientation y x (0, 3) (0, 23) (21, 0) (1, 0) (b) See Figure 75. Since the motion begins at the point 1, 0 , ( ) we want x 1 = and y 0 = when t 0. = The equation is an ellipse, so begin by letting x t t y t t cos and 3sin ω ω ( ) ( ) ( ) ( ) = = for some constant .ω These parametric equations satisfy the equation of the ellipse. Furthermore, with this choice, when t 0 = we have x 1 = and y 0. = For the motion to be counterclockwise, the motion has to begin with the value of x decreasing and the value of y increasing as t increases. This requires that 0. ω > (Do you know why?) Finally, since 1 revolution requires 2 seconds, the period is 2 2, π ω = so . ω π = The parametric equations that satisfy the conditions stipulated are x t t y t t t cos 3sin 0 2 π π ( ) ( ) ( ) ( ) = = ≤ ≤ (4) Now Work PROBLEM 39 The Cycloid Suppose that a circle of radius a rolls along a horizontal line without slipping. As the circle rolls along the line, a point P on the circle will trace out a curve called a cycloid (see Figure 76). Deriving the equation of a cycloid in rectangular coordinates is difficult, but the task is relatively easy using parametric equations. Figure 76 Cycloid x O X A B Y C 2a a t y P Begin with a circle of radius a and take the fixed line on which the circle rolls as the x-axis. Let the origin be one of the points at which the point P comes in contact with the x-axis. Figure 76 shows the position of this point P after the circle has rolled a bit. The angle t (in radians) measures the angle through which the circle has rolled. Since we require no slippage, it follows that AP d O A Arc , ( ) = The length of arc AP is given by s r ,θ = where r a = and t θ = radians. Then at d O A , ( ) = θ = s r , where = r a and θ = t The x-coordinate of the point P is dOX dOA dXA at a t a t t , , , sin sin ( ) ( ) ( ) ( ) = − = − = − The y-coordinate of the point P is dOY dAC dBC a a t a t , , , cos 1 cos ( ) ( ) ( ) ( ) = − = − = − Exploration Type the following inputs into the Desmos graphing calculator. When prompted, choose to "add slider" for a. Set the min and max values for a to be 0 and 20. Hit the play button or drag the slider for a to animate the graph in Figure 76.
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