Whenever a plan for major streets has been adopted, the council, upon recommendation of the planning commission, is authorized and empowered to establish, regulate and limit and amend, by ordinance, building or setback lines on major streets, and to prohibit any new building being located within building or setback lines. When a plan for proposed major streets or other public improvements has been adopted, the council is authorized to prohibit any new building being located within the proposed site or right-of-way when the center line of the proposed street or the limits of the proposed sites have been carefully determined and are accurately delineated on maps approved by the planning commission and adopted by the council. The council shall provide for the method by which this section shall be administered and enforced and may provide for a board of adjustment with powers to modify or vary the regulations, in specific cases, in order that unwarranted hardship, which constitutes an unreasonable deprivation of use as distinguished from the mere grant of a privilege, may be avoided. If there is a board of zoning adjustment on October 13, 1963, that board shall be appointed to serve as the board of adjustment for the building line regulations. If there is no board of zoning adjustment, the personnel, length of terms, method of appointment and organization of the board of adjustment for the building line regulations shall be the same as now provided for municipal boards of zoning adjustment. The regulations of this section shall not be adopted, changed or amended until a public hearing has been held thereon as provided in section 89.360.