Find the geodesic curvature of the line $y=b$ in the hyperbolic plane with metric $\frac{dx^2+dy^2}{y^2}$

Question

I try this:

Let $\gamma(t)=(bt,b)$. Then $\gamma '=(b,0)$ and $||\gamma'||=\frac{1}{b^2}.b.b=1$ => $\gamma$ is parametrized by arc length.

$\gamma''=(0,0)$=>$\kappa=||\gamma''||=0$. Since $|\kappa|\geq|\kappa_g|,\; |\kappa_g|=0$

But I know that $|\kappa_g|$ should be 1, so where is my mistake?

EDIT: My question is specifically for the hyperbolic plane, I want to know how to solve it in these coordinates. Secondly, even if we are talking about the hyperbolic circle I find the solution quite complicated and something more understandable would be welcome.