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y x = a cos f
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(x, y) y = a sin f
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Cartesian-coordinate graph of a circle with radius a, centered at the origin We can let f = t to describe this circle as a pair of parametric equations
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Polar-coordinate graph of a circle with radius a, centered at the origin We can let f = t to describe this circle as a pair of parametric equations
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Polar-coordinate graph to equations Now let s convert the circle in the previous example to a pair of polar-form parametric equations Suppose the polar direction angle is f, and the polar radius is r The equation of a circle having radius a as shown in Fig 16-11 is
r=a Let s call the angle f our parameter t, just as we did in the xy-plane situation Then we can write the parametric equations of our circle as f=t and r=a
Are you confused
Does the above pair of parametric equations seem strange to you The second equation doesn t contain the parameter! That s not a problem in this situation The parameter has no effect because the polar radius r is always the same
Here s a challenge!
Suppose that we come across a pair of parametric equations similar to the one in the Cartesiancoordinate example above, except that the cosine and sine of the parameter are multiplied by different nonzero real-number constants a and b, like this: x = a cos t
From Graph to Equations
and y = b sin t What sort of curve should we expect to get if we graph the relation defined by this pair of parametric equations
Solution
We ve been told that a and b are both nonzero real numbers Therefore, we can divide the equations through by their respective constants to get x /a = cos t and y /b = sin t If we square both sides of both equations, we obtain (x /a)2 = cos2 t and (y /b)2 = sin2 t When we add these two equations, left-to-left and right-to-right, we obtain the new equation (x /a)2 + (y /b)2 = cos2 t + sin2 t From trigonometry, we remember that for any real number t, it s always true that cos2 t + sin2 t = 1 Therefore, the preceding equation can be rewritten as (x /a)2 + (y /b)2 = 1 Expanding the squared ratios on the left-hand side gives us x2 /a2 + y2/b2 = 1 which is the equation of an ellipse centered at the origin The horizontal (x-coordinate) semi-axis is a units wide, and the vertical (y-coordinate) semi-axis is b units high
Parametric Equations in Two-Space
Practice Exercises
This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 In Rectangular-coordinate graph: example 1 (Fig 16-4), the parametric equations are x = t2 and y = t3 Find an equation for this relation that expresses x in terms of y without the parameter t Then find an equation that expresses y in terms of x without the parameter t 2 Is the relation defined in your second answer to Problem 1 a function of x 3 In Polar-coordinate graph: example 2 (Fig 16-7), the parametric equations are q = t 1 and r = ln t Find an equation for this relation that expresses q in terms of r without the parameter t Then find an equation that expresses r in terms of q without the parameter t 4 Is the relation defined in your second answer to Problem 3 a function of q 5 In Cartesian-coordinate graph to equations (Fig 16-10), the parametric equations are x = a cos t and y = a sin t where a is a nonzero constant Find an equation for this relation in terms of x and y only, without the parameter t 6 Express the solution to Problem 5 as a relation in which x is the independent variable and y is the dependent variable You should end up with y alone on the left-hand side of the equals sign, and an expression containing x (but not y) on the right-hand side Is this relation a function of x 7 Suppose that we come across the pair of parametric equations x = sec t and y = tan t Find an equation for this relation in terms of x and y only, without the parameter t What sort of curve should we expect to get if we graph this relation in the Cartesian xy plane
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