I posted this problem a few weeks ago:

The figure shows a fixed circle with equation and a shrinking circle with radius and center the origin (in red). is the point , is the upper point of intersection of the two circles, and is the point of intersection of the line and the -axis. What happens to as shrinks, that is, as ?

And I also posted this applet, so you could investigate it yourself.

Now, I post the solution. Initially, I thought that the -coordinate of would go off to infinity or zero, I wasn’t sure which. The equation for is . Solving for the intersection of the two circles, , we find it has coordinates

Remembering that , we now find the equation of line in point-slope form.

Now, we seek to find the coordinates of , the -intercept of the line. Letting in the above equation, we solve for and find

We now take the limit of as .

If we try to evaluate this limit by plugging in 0, we get an indeterminant form 0/0. We can either use L’hopital’s Rule or evaluate it numerically. Either way, we find.

I have to say, this result surprised me. Like I said, I expected this limit to evaluate to 0 or infinity–but 4?? I had to get a bit more of an intuitive understanding, so I built that Geogebra applet.

Also, I was told afterward, by the student who brought this to me, that there’s a straight-forward geometric way of deriving the answer, based on the observation that point Q and point R and the point (2,0) (0,2) form an isosceles triangle for all values of r. The observation isn’t trivial. Can you prove it’s true? Once you do realize this fact, though, the above result is clear.