SECTION 10.7 Plane Curves and Parametric Equations 739 Figure 65 ( ) ( ) = = −∞< <∞ > x t a t y t a t t a cos sin 0 y x (0, a) (2a, 0) (a, 0) Now Work PROBLEMS 7 AND 19 Let’s analyze the plane curve in Example 3 further. The domain of each parametric equation is t . −∞< <∞ So, the graph in Figure 65 is repeated each time that t increases by 2 .π If we wanted the curve to consist of exactly 1 revolution in the counterclockwise direction, we could write π ( ) ( ) = = ≤ ≤ x t a t y t a t t cos sin 0 2 This curve starts at t 0, = the point a, 0 , ( ) proceeds counter-clockwise around the circle, and ends at t 2 ,π = also the point a, 0 . ( ) If we wanted the curve to consist of exactly three revolutions in the counterclockwise direction, we could write π π ( ) ( ) = = − ≤ ≤ x t a t y t a t t cos sin 2 4 or x t a t y t a t t cos sin 0 6π ( ) ( ) = = ≤ ≤ or x t a t y t a t t cos sin 2 8 π π ( ) ( ) = = ≤ ≤ Describing Parametric Equations Find rectangular equations for the following plane curves defined by parametric equations. Graph each curve. (a) x t a t y t a t t a cos sin 0 0 π ( ) ( ) = = ≤ ≤ > (b) x t a t y t a t t a sin cos 0 0 π ( ) ( ) = − = − ≤ ≤ > Solution EXAMPLE 4 Figure 66 x t a t y t a t t a cos sin 0 0 π ( ) ( ) = = ≤ ≤ > y x (0, a) (2a, 0) (a, 0) (a) Eliminate the parameter t using a Pythagorean Identity. t t x a y a x y a cos sin 1 1 2 2 2 2 2 2 2 ( ) ( ) + = + = + = The plane curve defined by these parametric equations lies on a circle with radius a and center at 0, 0 . ( ) The curve begins at the point a, 0 , ( ) when t 0; = passes through the point a 0, , ( ) when t 2 ; π = and ends at the point a t , 0 , when .π ( ) − = The parametric equations define the upper semicircle of a circle of radius a with a counterclockwise orientation. See Figure 66. The rectangular equation is y a x a x a 2 2 = − − ≤ ≤ (b) Eliminate the parameter t using a Pythagorean Identity. t t x a y a x y a sin cos 1 1 2 2 2 2 2 2 2 ( ) ( ) + = − + − = + = The plane curve is a circle with center at 0, 0 ( ) and radius a. As the parameter t increases, say from t a 0 the point , 0 [ ] ( ) = to t a 2 the point 0, π [ ] ( ) = to t π = a the point , 0 , [ ] ( ) − the corresponding points are traced in a counterclockwise direction around the circle. The orientation is as indicated in Figure 65. (continued)
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