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At a Tangent
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1. Consider 31 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?
If there are 31 tangent lines, then there are 62 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?
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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