equation of a locus

Question

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

 

 

Answer 1

Let the moving point P be (x, y)

remember that slope m of line joining points (x1, y1) and (x2, y2) is m = (y1-y2)/(x1-x2)

slope of line joining point P(x, y) to A(1, 3) is (y-3)/(x-1)

slope of line joining point P(x, y) to B(3, 1) is (y-1)/(x-3)

We have been told that, slope of PA = 3 x slope of PB

(y-3)/(x-1) = 3 (y-1)/(x-3) ......multiply both sides by (x-1) (x-3)

(x-3)(y-3) = 3 (y-1)(x-1)

xy-3x-3y+9=3(xy-y-x+1)=3xy-3y-3x+3....subtract xy-3x-3y+9 from both sides

0 = 2xy -6 = 2(xy-3) = 0

xy -3 = 0 ...add 3 to both sides

xy = 3

y = 3/x is the equation of the locus