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Consider a circle of radius 4 centered at the origin in the XY-plane. Find a parametrization for...

Question:

Consider a circle of radius 4 centered at the origin in the XY-plane. Find a parametrization for the arc of this circle that lies in the fourth quadrant and has clockwise orientation. The point (4, 0) should correspond to t = 0 and the point (0, -4) should correspond to the right endpoint of the interval of t values.

{eq}\overrightarrow{r} (t) = ........................ \ \ for \ \ \ 0 \leq t \leq ............................ {/eq}

Note: Use t as the parameter for all of your answer and write {eq}\overrightarrow{r} (t) {/eq} in the form {eq}< x(t), y(t)> {/eq} without using {eq}\overrightarrow{i} {/eq} and {eq}\overrightarrow{j} {/eq}. In order to have your answer checked for correctness, you must have answers in both answer blanks.

Parameter Equation:

We will find the parametric equation of x and y in terms of t where the region lies in the fourth quadrant and the radius of the circle is 2. The equation of the circle will be {eq}x^{2}+y^{2}=4 {/eq}. We will write use a trigonometric identity to get the original equation in terms of x and y.

Answer and Explanation:

The equation of the circle lies in the fourth quadrant. The circle has a radius of 2.

The equation is:

{eq}x^{2}+y^{2}=4 {/eq}

In the parametric form, the x and y coordinates will be:

{eq}x=2\cos t\\ y=2\sin t {/eq}

Here t will vary from:

{eq}0\leq t\leq \frac{\pi}{2}\\ \vec{r}(t)=(2\cos t, 2\sin t) \text~~{for}~` 0\leq t\leq \frac{\pi}{2} {/eq}


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