- #1

StephenPrivitera

- 363

- 0

_{o}|=r, where z_o is a fixed point and r is a positive number, represents a circle centered at z_o and with radius r. |z - z

_{1}|=k|z - z

_{2}|, where z_1 and z_2 are fixed points, also apparently represents a circle, except maybe in the case where k=1. Then we have a line, or a circle of infinite radius. So to find the radius of the circle for |z - z

_{1}|=k|z - z

_{2}|, I could try to rewrite the equation to fit |z - z

_{o}|=r. I did this, and I got a frightening answer. I shall attempt to show it here. The work is much too long and too tedious to write here in full form but I'll explain briefly. x is the x component of z, y is the y component of z, x_1 is the x component of z_1, y_1...y component of z_1, etc.

Square both sides, distribute the k^2, collect x's and y's on the left, complete the square to get (x-somthing)^2+(y-same thing)^2=some big mess

simplify the right hand side, rewrite in terms of z1 and z2 as much as possible, take the sqrt of each side,

Anyone who feels like trying this problem could verify my result/ show me a better way of doing it?

r=(k

^{2}-1)

^{-1}[squ][(k

^{4}-2k

^{2}+2)|z

_{1}|

^{2}+k

^{2}(2k

^{2}-1)|z

_{2}|

^{2}-2k

^{2}(x

_{1}x

_{2}+y

_{1}y

_{2})]