07 October, 2007

Circles and a line

Fix a circle whose center lies on the x-axis, and whose edge passes through the origin, as in the diagram below. Call the radius of this circle R.In the diagram above, we have R = 2.

Consider a circle whose center lies at the origin. You have different options for the radius, so you can play around.Not every circle intersects the blue (fixed) circle, but some do. Keep those and discard the rest.

The circles you've kept intersect the curve. Doodle a little: for each of the reddish circles, draw a line from the top through the point of intersection.Notice that, as the reddish circles grow smaller, the line grows shallower. The point where this line meets the x-axis moves further right. Call this point the intercept of that line.

How far to the right does the intercept go?

When I first saw this, I thought the answer obvious. As the reddish circle diminishes to nothingness, the line descends to the x-axis, and the intercept proceeds on its jolly way to the right without interference from anything. It was "intuitively obvious" from the diagrams:Boy, was I wrong!

Let r indicate the radius of one of those red circles (whose center lies at the origin). Using nothing harder than high school algebra, one can obtain a formula for the intercept:

x = √4R2 - r2 + 2R.

As r dimishes to nothing, r2 also diminishes to nothing. The values grow no larger than 4R. What then is the smallest number that limits the intercept?

Enter Calculus. A tool called the limit allows one simply to substitute r = 0 into the expression above. We thus have

x = √4R2 +2R = 4R

(recalling that in a circle the radius is positive, so R > 0).

In the pictures I've been using, R = 2, so the farthest right the intercept will go is x = 8.

I'm not sure why, but I find this positively remarkable. It almost seems like a morality tale with several points.

(1) Intuition, despite its good reputation, is very often spectacularly wrong. Drawing a few pictures led me to the wrong conclusion; doing some algebra showed it.

(2) Had I drawn more circles, or simply been lucky enough to choose some "good" circles from the get-go, the resulting diagrams might have given me the right intuition after all. Then again, they might not.

(Here, "good" means "fairly small". The smaller circles will all generate intercepts very close to 4R.)

I also have questions. Why 4? What if the blue circle weren't fixed on the x-axis, but allowed to float anywhere in the plane? Who thought this up, and why? Why do I find this problem so beautiful? I suspect most people would have stopped long ago.

Note: This was based on a problem in Stewart's Single-Variable Calculus, 6th Edition, published by Brooks-Cole. Stewart actually fixed R = 1, but I suspect he based this on a problem he found elsewhere that was more general.

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