A graph $G$ is $k$-edge geodetic graph if every edge of $G$ lies in at least one geodesic of length $k$. We studied some basic properties of $k$-edge geodetic graphs. We investigated the $k$ edge-geodeticity of complete bipartite graph $K_{m,n}$ and provide the minimum number of largest fixed order path that can cover $K_{m,n}$. We also studied the $k$-edge geodeticity of tree and the product graphs like Cartesian product, Strong product, Corona product, and provide the bounds for the minimum number of the largest fixed order path that can cover the graph.