Find the equation of the locus of a moving point such that the slope of the line joining the point to A(1,3) is three times the slope of the line joining the point to B(3,1).

Answer 1

let the point be P(x,y)
slope AP = (y-3)/(x-1)
slope BP = (y-1)/(x-3)
(y-3)/(x-1) = 3(y-1)/(x-3)
xy - 3y - 3x + 9 = 3xy -3y - 3x + 3
2xy = 6

xy = 3

Test for reasonableness :
I know P(12, 1/4) lies on it

slope AP = (1/4 - 3)/(12-1)
= -1/4
slope BP = (1/4 - 1)/(12-3)
= -1/12

and -1/4 = 3(-1/12) , as needed