Rewrite the equation $(x-a)^2 + (y-b)^2 = r^2$ to make $y$ a function of $x$

Question

I'm trying hard to figure out how

$(x-a)^2 + (y-b)^2 = r^2$

can be written as

$y = b + \sqrt{r^2 - (x-a)^2}$.

My book says that

you’ll want to have $y$ as a function of $x$.

Answer 1

$$(x-a)^2+(y-b)^2=r^2$$ implies: $$(y-b)^2=r^2-(x-a)^2$$ Taking the square root of both sides then yields: $$y-b=\sqrt{r^2-(x-a)^2}$$ and adding $b$ to both sides of the equation: $$y=b+\sqrt{r^2-(x-a)^2}$$

Note that when we took the square root, we could have also taken the negative square root, since $x^2=(-x)^2$. So we could also have written: $$y=b-\sqrt{r^2-(x-a)^2}$$