Expert: Steve Holleran - 4/30/2007

Question
Hello Steve,
I am having trouble describing the locus of a point. Can you help? For example from the equation xÙ2  + yÙ2 - 2x + 7y + 2 = 0 How would i describe the locus of this point which moves so as to trace out the path represented by this equation?

Answer
Hi John,

I'm hoping I interpret your equation correctly.  You want to group the x and y terms on the left and complete the square in each:

 x^2 - 2x + y^2 + 7y = -2

 x^2 - 2x + 1  + y^2 + 7y + 49/4 = -2 + 1 + 49/4

(To complete the square, remember to take half the coefficient of the linear term and square it.  Then remember to balance by adding to both sides).

This becomes

   (x - 1)^2  + (y + 7/2)^2 = 45/4

This is the equation of a circle with center at (1, -7/2) and a radius of sqrt(45/4).

Hope this helps you out.

Steve Holleran