Show that the only curves whose curvature is fixed constant $k$ are circles of radius $1/k$.

Question

Show that the only curves whose curvature is fixed constant $k$ are circles of radius $1/k$.

The $r$ should probably be $k$. My question is what is done with the constants?

Answer 1

When you integrate $\int_0^x y'(u) ~du$, you get $y(x) - y(0)$; the "constant" is $y(0)$. In your case, the setup tells you what $y(0)$ is. Read the "choose coordinates" sentence again carefully.