Question
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).
Answers
Reiny
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
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