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2021-08-20

Give two pairs of parametric equations that generate a circle centered at the origin with radius 6.

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Given: A circle centered at the origin and radius 6.
To find: the two pairs of parametric equations
Concept used: in polar coordinates a point in the plane is identified by a pair of numbers(r, theta), where r and theta both are coordinate of the parametric system
Explanation: Let there is  a point (x,y)which belong to the circle which centered at origin and have radius of 6 unit.
${x}^{2}+{y}^{2}={6}^{2}$
${x}^{2}+{y}^{2}=36$
Now, using formula here as ${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$,
Using the above formula to create the parametric equation which are given as,
$x\left(t\right)=6\mathrm{sin}t$
and $y\left(t\right)=\mathrm{cos}t$ where t in $\left[0,2\pi \right]$
Simlarily, we can formulate the second parametric equation as,
$x\left(t\right)=6\mathrm{cos}t$
and $y\left(t\right)=\mathrm{sin}t$ where t in $\left[0,2\pi \right]$
Answer: The pair of parametric equations of a circle centered at origin and the radius is 6  are $x\left(t\right)=6\mathrm{sin}t$ and $y\left(t\right)=\mathrm{cos}t$
where t in $\left[0,2\pi \right]$ and $x\left(t\right)=6\mathrm{cos}t$
and $y\left(t\right)=\mathrm{sin}t$ where t in $\left[0,2\pi \right]$

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