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Math Help - Intersections Q

  1. #1
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    Intersections Q

    4 distinct parabolas and 3 distinct circles are drawn on a large sheet of paper. Find the maximum number of points of intersection of these curves. The answer is 78. Is there a better way than drawing it out :S
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  2. #2
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    No, I can't because I don't believe that 79 is correct! What reason do you have to believe that is the correct answer? I get a maximum of 42 points of intersection.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    No, I can't because I don't believe that 79 is correct! What reason do you have to believe that is the correct answer? I get a maximum of 42 points of intersection.

    This is confusing and I'm far from being sure, but I get 66:

    Every parabola can intersect with a circle at most at 4 points ==> we get 4*3*4 = 48 intersection points between the three circles and the 4 parabolas

    Every two parabolas can intersect in 2 different ponts ==> another 2*(1+2+3) = 12 points

    Every circle can intersect another circle at two points ==> 6 points more

    Sum them all you get 66 points...or I'm wrong, of course.

    Tonio
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  4. #4
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    Quote Originally Posted by vuze88 View Post
    4 distinct parabolas and 3 distinct circles are drawn on a large sheet of paper. Find the maximum number of points of intersection of these curves. The answer is 78. Is there a better way than drawing it out :S
    What does "distinct parabola" & "distinct circle" mean in this case?

    Any parabola perpendicular (or not oriented parallel) will have 4 intersections with any other parabola.

    Parabola1 & Parabola2 will have 4 intersections.
    P1&P3, P1&P4, P2&P3, P2&P4, P3&P4
    Intersections of all Parabolas = 24

    Each Circle and Parabola can have 4 intersection points.
    Curve1 & Parabola1 will have 4 intersections.
    c1&p2, c1&p3, c1&p4
    Intersections of all Parabolas & 1 circle = 16
    Intersections of all 3 circles and all 4 parabola = 48

    Intersection of circle1 & circle 2 = 2 points
    c1&c3, c2&c3
    Intersections of circles = 6

    Total intersections: 24 + 48 + 6 = 78

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