Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $λ$ be an arbitrary integral weight. We denote by $\mathbb{B}(λ)$ the crystal of all Lakshmibai-Seshadri paths of shape $λ$. Let $V(λ)$ be the extremal weight module of extremal weight $λ$ generated by the (cyclic) extremal weight vector $v_λ$ of weight $λ$, and let $\mathcal{B}(λ)$ be the crystal basis of $V(λ)$ with $u_λ\in \mathcal{B}(λ)$ the element corresponding to $v_λ$. We prove that the connected component $\mathcal{B}_0(λ)$ of $\mathcal{B}(λ)$ containing $u_λ$ is isomorphic, as a crystal, to the connected component $\mathbb{B}_0(λ)$ of $\mathbb{B}(λ)$ containing the straight line $π_λ$. Furthermore, we prove that if $λ$ satisfies a special condition, then the crystal basis $\mathcal{B}(λ)$ is isomorphic, as a crystal, to the crystal $\mathbb{B}(λ)$. As an application of these results, we obtain an algorithm for computing the number of elements of weight $μ$ in $\mathcal{B}(Λ_1-Λ_2)$, where $Λ_1, Λ_2$ are the fundamental weights, in the case that $\mathfrak{g}$ is symmetric.