Expand the squares on the left-hand sides and subtract the two equations. The $x_a'^2$ and $y_a'^2$ terms will cancel out, and what is left is a linear equation in $x_a'$ and $y_a'$ which you can use to express $y_a'$ in terms of $x_a'$ (or vice versa -- one of these may be impossible depending on the constants).
Stick the expression for $y_a'$ into one of the original equations and simplify. The result is a quadratic equation in $x_a'$ which you can solve with the quadratic formula. Finally use the linear equation again to find the corresponding value of $y_'a$.
If you try to unfold this as a single closed formula in $x_f$, $y_f$, $r$ $x_a$, $y_a$, and $\alpha$, the resulting formula is going to be huge alright. But if you give names to some of the intermediate values and split the whole thing into steps, each step is individually simple enough.