Hints and solutions to
Problem of the day for Friday, November 8
What is the maximum number of points of intersection when two
distinct circles and three distinct lines intersect each other?
- This definitely requires a picture!
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Or does it? Well, the picture above indicates a lot of the possibilities.
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But we can be pretty systematic about this:
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Two lines can intersect in one point. So with
three lines, we can
have at most three points of intersection.
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Now let's think about the circles.Two circles can intersect in at most
two points,
and a line can intersect a circle in at most two points.
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So let's count: each of the three lines intersects the other two lines
once, and
the circles twice - that makes 6 intersection points for each of the
three lines, or
18 intersection points.
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Now for the circles. Each circle intersects the other in two points, and
each
of the three lines
in two points. So each circle has 8 intersection points That makes 16
intersection points
between the two circles.
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So that's 18 + 16 = 34 intersection points. But like last Friday,
we've counted
each intersection twice (two things intersect).
So you can have at most 17
intersection
points in all.