SOLUTION: Find the equation of the locus of a point which moves so that its distance from the point (1,-1) is three times its distance from the line y = 3.

Write the equation for distances, as you read the problem

     = 3*|y-3|, 

or

     = 3*|y-3|.


Square both sides

    (x-1)^2 + (y+1)^2 = 9*(y-3)^2.


Simplify and reduce to the standard conic section equation

    (x-1)^2 +  y^2 + 2y + 1 = 9y^2 - 54y + 81

    (x-1)^2 - 8y^2 + 56y + 1  = 81

    (x-1)^2 - 8(y^2 - 7y) + 1 = 81

    (x-1)^2 - 8(y^2 - 2*3.5y + 3.5^2) + 8*3.5^2 + 1 = 81

    (x-1)^2 - 8(y-3.5)^2 = 81 - 8*3.5^2 - 1

    (x-1)^2 - 8(y-3.5)^2 = - 18

    8(y-3.5)^2 - (x-1)^2 = 18


     -  = 1.


The last equation describes a hyperbola with the center at the point (1,3.5),

vertical real semi-axis of the length   =   (tranverse semi-axis),

and horizontal imaginary semi-axis of the length   = .


The hyperbola is open vertically up and down.