Our primary interests in this talk are the Lagrangians spheres in a symplectic 4-manifold M with b+ = 1. We find smooth embedded connected symplectic surfaces intersecting a given Lagrangian sphere minimally. This result turns out very useful in both the existence and uniqueness problems of Lagrangian spheres. We also discuss its implication on the homological action of diffeomorphisms and symplectomorphisms of a rational manifold.