Parametrization of line segment on manifold

Question

So I have a regular surface $S$ parametrized by $x(u,v)=(u,v,\sqrt(u^2 + v^2 +1))$ for $x,y\in R^2$ and I would like to find the arc length of the segment connecting two points $A=(0,0,1)$ and $B=(0,a,\sqrt(a^2 +1))$. From what I understand I can do this using the first fundamental form and a parametrization of the line segment of the form $a(t)=x(u(t),v(t))$. I have calculated the coefficients of the fundamental form and have a parametrization of the segment $\gamma (t)=(1-t)*A+t*B$ but am not sure how I should go about finding the parametrization in the required form. Any help is appreciated!