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Homework Help: Ambiguous Problem involving Common Tangents to circles

  1. Aug 19, 2009 #1
    In my math exam , this question had appeared :

    http://img33.imageshack.us/img33/6233/44467585.jpg [Broken]
    (You can click on the link to see the question)

    I'm having confusion as to what the answer to the question is.

    I feel that the correct answer to the question would be 2 Direct common tangents , as a common tangent is a tangent is a tangent that is a common tangent to all the circles under consideration.(according to me)

    However , my teacher feels otherwise and says that there are actually 3 common tangents as there is also a Transverse common tangent to the pair of touching circles in the diagram. Here is a diagram of his view:
    http://img194.imageshack.us/img194/1092/60396510.jpg [Broken]

    If we take my teachers view into consideration , then there are 2 transverse common tangents to the first and the last circles too. Hence the total number of tangents goes up to 5. Here is that diagram :
    http://img20.imageshack.us/img20/9199/20409914.jpg [Broken]

    However , my teacher does not agree with this. He says that the TCTs cease to be TCTs as they intersect the middle circle.
    My main question here is with regard to my teachers rather fishy statement "The TCT ceases to be a tangent as it is the secant of yet another circle" . Is he correct?

    If you think of it in another way and consider separate DCTs for pairs for circles, instead of a single DCT , then the number of tangents can go upto 11.

    Also , no where in the question is it mentioned that the centers of the circles are collinear. Hence the no of tangents can be 0 also.

    So what is the correct answer to this ambiguous question?
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 19, 2009 #2

    Ben Niehoff

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    Who wrote the question? Your teacher, or some other agency?

    Your teacher's answer is inconsistent. Either he should ask for lines which are tangent to all three circles, or he should allow any line which is a tangent to any pair of circles. Crossing the center circle does not cause the line to cease to be a tangent to the other two.

    Whichever way the question is supposed to be interpreted, it should be made clearer.
  4. Aug 20, 2009 #3


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    Even if we accept the teacher's definition for "common tangent", three is not the answer. Imagine a point above the circles, say directly above the center of the middle circle and outside all three circles. You can draw two tangents one to the leftmost and one to the rightmost circle that clearly do not intersect any of the circles. You can do the same with a point below the circles. So, the number of tangents that meet the teacher's definition is seven.
  5. Aug 20, 2009 #4
    Read this article.

    In trivial case where their centers are collinear, there are 5 tangents.

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