given a circle of radius $r$, centered at the origin, and a point $P_1 = (x_1,y_1)$ inside the circle ($x_1^2 + y_1^2 < r^2$), find a point $P_2 = (x_2,y_2)$ outside of it ($x_2^2 + y_2^2 > r^2$) such that, the ratio of the distances from any point $P = (x,y)$ of the circumference (x^2 + y^2 = r^2) is constant: \begin{equation*} \frac{d(P,P_1)}{d(P,P_2)} = k, \;\;\;\; k \in \mathbb{R}^+ \end{equation*} what i want to know is if $P_2$ exists, if its unique and if a formula can be found to determine it from the coordinates of $P_1$ and the value of $r$.
regarding the motivation for this problem, i was thinking about this as it relates to poles and zeros of Z transfer functions in all pass filters, i know and have confirmed that in the case of the unit circle, with $P_1 = re^{i\theta}$ (0 < r < 1) then choosing $P_2 = \frac{1}{P_1^*} = \frac{1}{r}e^{i\theta}$ satisfies the condition and k = r. with that i have the existence and formula part of my question answered but i wish to know if there is more than one possible choice of $P_2$ and also how to come up with the solution in the first place.