738 CHAPTER 10 Analytic Geometry 3 Find a Rectangular Equation for a Plane Curve Defined Parametrically The plane curve given in Examples 1 and 2 should look familiar. To identify it accurately, find the corresponding rectangular equation by eliminating the parameter t from the parametric equations given in Example 1: x t t y t t t 3 2 2 2 2 ( ) ( ) = = − ≤ ≤ Solve for t in y t2 , = obtaining t y 2 , = and substitute this expression in the other equation to get x t y y 3 3 2 3 4 2 2 2 ( ) = = = ↑ = t y 2 This equation, x y3 4 , 2 = is the equation of a parabola with vertex at 0, 0 ( ) and axis of symmetry along the x-axis. We refer to this equation as the rectangular equation of the curve to distinguish it from the parametric equations. Note that the plane curve defined by equation (1) and shown in Figure 64 is only a part of the parabola x y3 4 . 2 = The graph of the rectangular equation obtained by eliminating the parameter will, in general, contain more points than the original plane curve. Care must therefore be taken when a plane curve is graphed after eliminating the parameter. Even so, eliminating the parameter t from the parametric equations to identify a plane curve accurately is sometimes a better approach than plotting points. Exploration Graph x y y x y x 3 4 4 3 and 4 3 2 1 2 = = =− ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ using a graphing utility with = Xmin 0, = =− = X Y Y max 15, min 5, max 5. Compare the graph with Figure 64. Why do the graphs differ? Finding the Rectangular Equation of a Plane Curve Defined Parametrically Find the rectangular equation of the plane curve whose parametric equations are x t a t y t a t t cos sin ( ) ( ) = = −∞< <∞ where a 0 > is a constant. Graph the plane curve, and indicate its orientation. EXAMPLE 3 Solution The presence of sines and cosines in the parametric equations suggests using a Pythagorean Identity. In fact, since t x a t y a cos and sin = = this means that x a y a x y a 1 2 2 2 2 2 ( ) ( ) + = + = + = t t cos sin 1 2 2 Exploration Graph the following parametric equations using a graphing utility with =− = X X min 1, max 15, Y Y T min 5, max 5, and step 0.1: =− = = 1. x t t y t t t 3 4 , , 4 4 2 ( ) ( ) = = − ≤ ≤ 2. x t t t y t t t 3 12 12, 2 4, 4 0 2 ( ) ( ) = + + = + − ≤ ≤ 3. x t t y t t t 3 , 2 , 8 8 2 3 3 ( ) ( ) = = − ≤ ≤ Compare these graphs to the graph in Figure 64. Conclude that parametric equations defining a curve are not unique; that is, different parametric equations can represent the same graph.
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