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equation of a locus

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)  

 

 

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1 Answer

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