Question

the variable point P(x, y) moves so that it is the same distance from the points (1,6) and (3, 2). The equation of the locus of P may be obtained by?

Answer 1

let the point be P(x,y)
so
√( (x-1)^2 + (y-6)^2 ) = √( (x-3)^2 + (y-2)^2 )
square both sides, and expand

x^2 - 2x + 1 + y^2 - 12y + 36 = x^2 - 6x + 9 + y^2 - 4y + 4

4x - 8y = -24
x - 2y = -6

Notice that this is the equation of the right-bisector of the line joining the two points

other way of doing it:
slope of given line = (6-2)/(1-3) = -2
so the slope of the right-bisector is 1/2
midpoint for given points = (2,4)
equation:
y-4 = (1/2)(x-2)
2y - 8 = x - 2
x - 2y = -6 , same as above