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We typically see hyperbolas drawn the "nice" way. Namely, they are oriented with the arms "opening up" straight up or down, or at 45 degrees. But, in general, they can be at any "angle".

Question: Consider two hyperbolas. Each has two arms of course. Each hyperbola can be at any "angle" as noted above. Now, at how many points can the hyperbolas cross, having the same x,y value.

I think the answer is zero, one, two, three, or four depending on the equations, but I cannot prove it. If I asked the question of number of solutions for a straight line and a circle, it's clear the answer is zero, one, or two.

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  • $\begingroup$ My intuition says either 4, 2 or 0, but I could be wrong. Consider working with two parabolas first, it may clue you in on an answer. $\endgroup$ – Jeffrey L. Jun 3 '15 at 14:46
  • $\begingroup$ If "cross" includes tangency (and I suppose it must, for 1 or 3 "crossings"), than I suspect you're right that all of those values are possible. I'll try to come up with some pictures, at least. $\endgroup$ – pjs36 Jun 3 '15 at 16:09
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You are right: see pictures below.

enter image description here

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