Prove that the homothety $z\mapsto\lambda z $ (for$\lambda\gt0$)is a hyperbolic isometry of $\mathbb H^2$

Question

Prove that the homothety $z\mapsto\lambda z $ (for$\lambda\gt0$)is a hyperbolic isometry of $\mathbb H^2$

I'm unsure what this problem is even asking, how do I prove hyperbolic isometry

Answer 1

Consider $z=x+iy$, the hyperbolic metric at $z$ is ${{uu'+vv'}\over y^2}$ where $(u,v)$ and $(u',v')$ are tangent to $z$.

Write $h_\lambda=\lambda z$, $(h^*_\lambda)<>_z=<\lambda(u,v),\lambda(u',v')>_{h(z)}={{\lambda^2(uu'+vv')}\over{\lambda^2y^2}}=<u,v>_z$.