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[Community Question] Geometry: The limit about the line connecting the intersection of a circle and the $y$-axis and the intersection of the shrinking circle and a fixed circle

One of our user asked:

There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$, $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$-axis. What happens to $R$ as $C_2$ shrinks, that is, as $r \to 0^+$? (The figure is made with GeoGebra)

The figure

In order to solve this problem, I made a script using GeoGebra in which the circle $C_2$ is a dynamic one whose radius $r$ can be adjusted with a slider. As I set $r \to 0^+$, the figure seems to suggest that $R \to (4,0)$. In particular, this is the state with $r = 0.001$, in which $R$ is reported to be $(3.9999997523053,0)$:

the state with r = 0.001

However, I would like to find out a way to prove (or disprove, though unlikely) my guess that $$\lim_{r \to 0} R = (4,0).$$ But I have little idea. Any help would be appreciated.


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