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At a Tangent

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1. Consider 21 lines drawn at a tangent to an ellipse. The lines break up the plane into several regions, some bounded (i.e., enclosed) and others unbounded. How many unbounded regions are there?
Answer: 42


If there are 21 tangent lines, then there are 42 unbounded regions.

It is not very difficult to visualize that each tangent line drawn creates two unbounded regions.

The animation shows 3 tangent lines. The unbounded regions are in yellow and the bounded regions in green.
For 3 tangent lines, there are 6 unbounded regions.

Food for thought:

How do you arrive at the solution to such a problem?
By induction?!
Try to solve the problem for 1 tangent line, 2 tangent lines, and so on. It is easy to then generalize.

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puzzle : image for tangents

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