# Give two pairs of parametric equations that generate a circle centered at the origin with

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.
$$\displaystyle{x}^{{2}}+{y}^{{2}}={6}^{{2}}$$
$$\displaystyle{x}^{{2}}+{y}^{{2}}={36}$$
Now, using formula here as $$\displaystyle{{\sin}^{{2}}\theta}+{{\cos}^{{2}}\theta}={1}$$,
Using the above formula to create the parametric equation which are given as,
$$\displaystyle{x}{\left({t}\right)}={6}{\sin{{t}}}$$
and $$\displaystyle{y}{\left({t}\right)}={\cos{{t}}}$$ where t in $$\displaystyle{\left[{0},{2}\pi\right]}$$
Simlarily, we can formulate the second parametric equation as,
$$\displaystyle{x}{\left({t}\right)}={6}{\cos{{t}}}$$
and $$\displaystyle{y}{\left({t}\right)}={\sin{{t}}}$$ where t in $$\displaystyle{\left[{0},{2}\pi\right]}$$
Answer: The pair of parametric equations of a circle centered at origin and the radius is 6  are $$\displaystyle{x}{\left({t}\right)}={6}{\sin{{t}}}$$ and $$\displaystyle{y}{\left({t}\right)}={\cos{{t}}}$$
where t in $$\displaystyle{\left[{0},{2}\pi\right]}$$ and $$\displaystyle{x}{\left({t}\right)}={6}{\cos{{t}}}$$
and $$\displaystyle{y}{\left({t}\right)}={\sin{{t}}}$$ where t in $$\displaystyle{\left[{0},{2}\pi\right]}$$