Placing geometric objects along a line

Question

Say we've got a function such as $f(x)=x \cdot \sin(x)$ and we want to place a line of length $L$ on the graph that has its start and endpoints located on the line generated by $f(x)$. Assuming that we choose the starting point $(x_1, f(x_1))$, is there a way to analytically (not numerically) find the coordinate $(x_2, f(x_2))$ such that $\sqrt{(x_1 - x_2)^2 + (f(x_1) - f(x_2))^2} = L$?