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At a Tangent
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1. Consider 8 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 8 tangent lines, then there are 16 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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