Question

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ \u003c t \u003c ∞. x = 1 + 3 cos t, y = 2 + 3 sin t; 0 ≤ t \u003c 2π

Accepted Answer

Identify the parametric equations given: $$x = 1 + 3 \cos t$$ and $$y = 2 + 3 \sin t$$, with $$0 \leq t \u003c 2\pi. $$Recall the Pythagorean identity: $$\cos^2$ t + \sin^2$ t = 1. $$This will help eliminate the parameter $$t by $$expressing $$\cos t$$ and $$\sin t in $$terms of $$x$$ and $$y. $$Rewrite $$\cos t$$ and $$\sin t$$ from the parametric equations: $$\cos t = \frac{x - 1}{3}$$ and $$\sin t = \frac{y - 2}{3}. $$Substitute these expressions into the Pythagorean identity to get the rectangular equation: $$\left(\frac{x - 1}{3}\$$right)^2$$ + \left(\frac{y - 2}{3}\$$right)^2$$ = 1. $$Recognize that this equation represents a circle centered at $$(1, 2)$$ with radius $$3. To $$sketch the curve, draw this circle and use the parameter interval $$0 \leq t \u003c 2\pi to $$determine the orientation, which corresponds to increasing $$t$$ moving counterclockwise starting from the point $$(4, 2).$$

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Concetti chiave

Parametric Equations

Eliminating the Parameter

Orientation and Sketching of Parametric Curves