Let (x,y) is the point of the locus (of the set of points).


Its distance from the point (3,2) is  .

Its distance from the point (-1,4) is   = .


From the condition,

     = .


Square both sides

    (x-3)^2 + (y-2)^2 = 4*((x+1)^2 + (y-4)^2).


FOIL

    x^2 - 6x + 9 + y^2 - 4y + 4 = 4x^2 + 8x + 4 + 4y^2 - 32y + 64

    3x^2 + 14x + 3y^2 - 28y = -51

    x^2 +  + y^2 -  = - 17

    x^2 +  +  + y^2 -  +  = -17 +  + 

    (x+14/6)^2 + (y-28/6)^2 = -17 +  = -17 +  = .


Answer.  This locus (= this set of points) is the circle with the equation

          +  = .

         The circle is centered at (,)  and has the radius  of    units.