Given the metric of the Poincaré upper half-plane model

$$(ds)^2 = \frac{(dx)^2 + (dy)^2}{y^2}$$

and two known points $(x_1, y_1)$ and $(x_2, y_2)$ in the corresponding hyperbolic space $\mathbb{H} = \{ (x ,y) \mid y > 0; x, y \in \mathbb{R} \}$, can Mathematica give me the distance between those points without resorting to the distance formula

$$d((x_1, y_1), (x_2, y_2)) = \operatorname{arcosh} \left( 1 + \frac{ {(x_2 - x_1)}^2 + {(y_2 - y_1)}^2 }{ 2 y_1 y_2 } \right)$$

The problem I have is, for some $t \in [0,1]$, I need a distance formula for a hyperbolic metric

$$\frac{e^{2ty}(dx)^2+(dy)^2}{y^2}$$

but this is difficult to obtain, since I need to solve the geodesic equations. All I really need is a set of distances between some Poisson points drawn from $\mathbb{H}$ with this new metric, in order to form a random graph.

Is there a way of doing this using e.g. a built in function(s) in Mathematica?