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Obviously, the possible numbers if intersections for, one branch of a hyperbola and one branch of another hyperbola, are: $0, 1, 2, 4$ (check the example here). Is it possible for $3$?

If the two branches share the same focus, what's the maximum number of intersections?

EditPiAf
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baisong
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    i would - absolutely in an informal way - say that if you shift the green hyperbola to the right eventualy will be tangent to the purple one - leading to three intersection points without multiplicity – JayTuma Aug 14 '18 at 10:05
  • Is it even possible to shift from 2 to 4 without ever passing through 3? – Ivan Neretin Aug 14 '18 at 12:23
  • @JayTuma apperently correct, see the example graph, when we slowly move the green one to the right, the upper two intersection points will goes smoothly closed to each other and finally they will merge in one, then we get 3 intersections. Is this proof sound? – baisong Aug 14 '18 at 14:00

1 Answers1

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The only way to get three intersection points is if one of them is a "double" intersection (i.e. a point where the two curves are tangent to one another).

To see why ...

You can write one curve in parametric form $\mathbf{X} = \mathbf{X}(t)$, using rational quadratic functions, and the other in implicit form $f(\mathbf{X}) =0$, then intersections occur at real values of $t$ that satisfy $f(\mathbf{X}(t)) =0$. This leads to a polynomial equation of degree $4$. This equation generically has either 0, 2, or 4 real roots. If there are two real roots, and they are equal, then the two curves are tangent at some point, and you might say that there is only one intersection. If there are four real roots, and two of them coincide, then you might say that there are three intersections.

bubba
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  • Thanks, sounds like so although I don't understand the full story. How about the last question,

    If the two branches share the same focus, what's the maximum number of intersections?

     – baisong Aug 14 '18 at 14:03