Converting $y = -\sqrt{1 - x^2} + 2$ to polar coordinates

Question

Question: Convert $y = -\sqrt{1 - x^2} + 2$ to polar coordinates:

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What I have done

$$ y = -\sqrt{1 - x^2} + 2 $$

$$ 2-y = \sqrt{1 - x^2} $$

$$ (2-y)^2 = (1-x^2) $$

$$ x^2 + y^2 -4y + 3 = 0 $$

$$ x^2 + y^2 -4y = -3 $$

$$ x^2 + (y^2 -4y + 4) = -3 + 4 $$

$$ x^2 + (y-2)^2 = 1 $$

Now I am stuck where do I continue from here?

Answer 1

Use $$\cos^2 z+\sin^2 z=1$$ to write your variables $x,y$ in terms of $z$.

Answer 2

In 3rd to last line, substitute $x^2+y^2=r^2$ and $y=r\sin\theta$ and that should be enough.