finding parametric equations from a rectangular equation

Question

Find the parametric equations for $x^2-4x+y^2-2y+5=2$, and graph. Hint: Complete some squares. I have completed squares and gotten $(x-2)^2+4=-(y+1)^2-2$ but I am confused with how to proceed. I know it will be a circle but how do I change this into parametric equations and graph points? They want the final form to be H(t)=(f(t), g(t))

Answer 1

hint: $$(x-2)^2 + (y+1)^2 = (\sqrt{2})^2\to x-2 = \sqrt{2}\cos \theta,y + 1 = \sqrt{2}\sin \theta$$.

If you have $\triangle^2 + \square^2 = A$, you can write $\left(\dfrac{\triangle}{\sqrt{A}}\right)^2+\left(\dfrac{\square}{\sqrt{A}}\right)^2 = 1$, then you can always set $\dfrac{\triangle}{\sqrt{A}} = \cos \theta$, and $\dfrac{\square}{\sqrt{A}} = \sin \theta$, for some angle $\theta \in [0,2\pi]$