Meeting probability

There is a grid of roads (MxN). Two people, one at A (0,0) and the other at B (N,M), start to walk from A to B and B to A respectively at the same time. They both walk the shortest path to their destination. For instance, if you come from B to A, you must always move from East to West (left) or North to South (down). The man at B walks faster than the one at A by a factor of k. (v_B=k*v_A where k>1) If they walk in any possible direction equally likely on a 19x23 grid, what is the probability that they meet each other on their way if k is equal to 4/3 (1) or 6/5 (2)? Answer Format : P₁/Q₁,P₂/Q₂ Example: Answer=5/14,5/28 for k=2 and k=3/2 on 3x3 grid. The following shows a possible meeting for k=2: