Question 71320
Here two things are at work :-
1) Distance between two points (a,b) & (x,y) =  ((x-a)^2 + (y-b)^2)^0.5
2) To get the locus we convert the wordings given in the problem in the form   
   of an equation.

Now, to solve the problem let's denote a general point by (x,y) so from given condition:-
  
  ((x-3)^2 + (y-4)^2)^0.5 = 4((x+5)^2 + (y-2)^2)^0.5
  ((x-3)^2 + (y-4)^2)     = 16((x+5)^2 + (y-2)^2)
  x^2+9-6x+y^2+16-8y = 16x^2+16*25+2*16*5x+y^2+4-4y
15x^2 + 166x + 4y + 379 = 0   This is the required equation of locus.

This graph is a parabola