$12.(4$ ?Points $)$ ?East Massachusetts Healthcare Institution is preparing to build a research center to serve its four sources of patients. The four hospital locations at coordinates and the volumes are provided.

a $.$ ?Regardless of the volume from each hospital, find the $$ feasible region $$ ?for the research center location. Hint: Feasible region is a geometric area where the center $-$ of $-$ gravity is located in $.$

b $.$ ?Calculate the volume $-$ weighted center $-$ of $-$ gravity for the research center location.