Negative edge weights in Dijkstra and Bellman Ford shortest path algorithms
Then why texts say that Dijkstra need all non negative edge weights?


Because it has to work for all cases, not only a few examples.


  I feel that it should be no -ve edge weight cycle reachable from source node.


Try proving that claim and see where it leads. If you're successful, great; if not, trying to prove the opposite will fix your intuition.

See also here for a high-level argument.


   The smallest example where Dijsktra fails has four nodes, no (directed) cycle at all, and no negative-value cycle: