Problem Statement:
Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ be two fixed points on xy plane and $R(\alpha , \beta)$ is a point such that $PR:QR=k , \ (k≠1)$ and locus of R for different values of k be curves $S_1,S_2, \dots$. Further let $P _1, P_2, \dots $ be circles such that $S_i $ is orthogonal to $P_j$ $\forall \ i,j$.
- Number of common tangents of $S_i$ and $S_j$ maybe: (i≠j)
- Number of common tangents of $P_i$ and $P_j$ maybe: (i≠j)
My attempt: I couldn't go beyond problem 1. For simplicity I assumed point $P(0,0) $ and $Q(x_2,0)$, and proved that if centres of say S1, and S2 are O1, O2 then $O_1O_2< r_1 -r_2$, where r1, r2 are respective radii. And thus, no common tangents.
How does one solve the second part? Also is there more rigorous solution of first?
Thanks.