Locus

let A be(a,b) and B be (c,d) and P be (h,k)

then

(h-a)² + (k-b)² - (h-c)² + (k-d)²= l(constant)

2h(c-a) + 2k(d-b) + a² +b² -c² -d² = l

put (h,k) --> (x,y)

then clearly it is a equation of straight line.