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 −∞ < t < ∞.

x = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π

Accepted Answer

Identify the given parametric equations: $$x = 2 + 4 \cos t$$ and $$y = -1 + 3 \sin t$$, with the parameter $$t in $$the interval $$0 \leq t \leq \pi. $$Express $$\cos t$$ and $$\sin t in $$terms of $$x$$ and $$y by $$isolating them from the parametric equations: $$\cos t = \frac{x - 2}{4}$$ and $$\sin t = \frac{y + 1}{3}. $$Use the Pythagorean identity $$\sin^2$ t + \cos^2$ t = 1 to $$eliminate the parameter $$t. $$Substitute the expressions for $$\cos t$$ and $$\sin t$$ into this identity: $$\left(\frac{x - 2}{4}\right)^2 + \left(\frac{y + 1}{3}\right)^2 = 1.$$ Recognize that this equation represents an ellipse centered at $$(2, -1)$$ with semi-major axis 4 along the $$x-$$direction and semi-minor axis 3 along the $$y-$$direction. To sketch the curve, plot this ellipse and indicate the orientation by considering the direction of increasing $$t$$ from $$0 to$$ $$\pi$$, which corresponds to moving from the point where $$t=0 ($$start) to $$t=\pi ($$end) along the ellipse.

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Concetti chiave

Parametric Equations

Eliminating the Parameter

Orientation and Interval of the Parameter