Given three points, can one construct a hyperbolic curve using classical geometric construction method?

Question

Using straight edge and compass method, can a hyperbolic curve be drawn through three given points?

Answer 1

Well, given five points, a conic section is determined and any number of desired points may be found by Pascal's Theorem. http://en.wikipedia.org/wiki/Pascal%27s_theorem

I don't see that three is enough...there is only one circle through three points but infinitely many ellipses