Why do we get the following radius from rotating the section of the graph about $y=12$?

Question

I understand the reasoning when we rotate the same graph about the $x$ axis, but when we rotate the graph about $y=12$, why does $r$ become $12 - 9/ x^2$ and $12 − (10 − x^2)$? Why wouldn't they become $9 / x^2 + 12$ and $10 - x^2 + 12$ ?

Graph + solution

Answer 1

Plug $x=1$ into the first, you have the point $(1,3)$. Its distance from $y=12$ is $9$, not $15$ !

Answer 2

For any fixed $x$, the center of the circle is at $12$, and you know that $9/x^2$ is on the circle itself, which means that the radius is equal to the distance between $12$ and $9/x^2$. Distances between numbers are found by subtracting.