  Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let
$\lambda$ be an arbitrary integral weight. We denote by $\mathbb{B}(\lambda)$
the crystal of all Lakshmibai-Seshadri paths of shape $\lambda$. Let
$V(\lambda)$ be the extremal weight module of extremal weight $\lambda$
generated by the (cyclic) extremal weight vector $v_\lambda$ of weight
$\lambda$, and let $\mathcal{B}(\lambda)$ be the crystal basis of $V(\lambda)$
with $u_\lambda \in \mathcal{B}(\lambda)$ the element corresponding to
$v_\lambda$. We prove that the connected component $\mathcal{B}_0(\lambda)$ of
$\mathcal{B}(\lambda)$ containing $u_\lambda$ is isomorphic, as a crystal, to
the connected component $\mathbb{B}_0(\lambda)$ of $\mathbb{B}(\lambda)$
containing the straight line $\pi_\lambda$. Furthermore, we prove that if
$\lambda$ satisfies a special condition, then the crystal basis
$\mathcal{B}(\lambda)$ is isomorphic, as a crystal, to the crystal
$\mathbb{B}(\lambda)$. As an application of these results, we obtain an
algorithm for computing the number of elements of weight $\mu$ in
$\mathcal{B}(\Lambda_1-\Lambda_2)$, where $\Lambda_1, \Lambda_2$ are the
fundamental weights, in the case that $\mathfrak{g}$ is symmetric.
