Given #x^2 + (y – 2)^2 = 4 # how do you derive a parametric equation?

Question

Charles Arocha · Accepted Answer

The parametric equations are #x=2costheta# and #y=2+2sintheta#

The equation represents a circle, center #(0,2)# and radius #r=2#

We use the following parametric equations

#x=rcostheta#

and #y-2=rsintheta#

Therefore,

#x^2+(y-2)^2=r^2cos^2theta+r^2sin^2theta=4#

So,

#r^2(cos^2theta+sin^2theta)=4#

#r=sqrt4=2#

As #cos^2theta+sin^2theta=1#

The parametric equations are

#x=2costheta# and #y=2+2sintheta#

Christopher Agresta · Answer

undefined To derive a parametric equation from the equation $x^2 + (y - 2)^2 = 4$, you can let $x = r \cos(\theta)$ and $y = r \sin(\theta)$, where $r$ represents the radius and $\theta$ represents the angle.

Substitute these expressions for $x$ and $y$ into the given equation and solve for $r$:
$$ (r \cos(\theta))^2 + ((r \sin(\theta)) - 2)^2 = 4 $$

Simplify the equation and solve for $r$.

Once you have found $r$ as a function of $\theta$, you can write the parametric equations:
$$ x = r \cos(\theta) $$
$$ y = r \sin(\theta) $$

where $r$ is the expression you found earlier in terms of $\theta$. This will give you the parametric equations representing the curve defined by the equation $x^2 + (y - 2)^2 = 4$.