I seem to have a lot of confusion in understanding the concept of asymptotes to a curve, particularly to hyperbola(it seems very difficult for me to visualize since there are two branches to a hyperbola)
Since as a result of my lack of basic understanding particularly in this chapter,I am unable to even phrase my doubts properly.Apologies in advance!
For the sake of simplicity,I am using a standard hyperbola x^2/a^2 - y^2/b^2 =1,to describe my doubts
i)I know that the condition for tangency of any line y=mx+c to a hyperbola is c^2=a^2m^2- b^2
If we take any arbitrary line tangent to one of the branch of the hyperbola,it seems to me that the tangent to the branch would always cut the other branch.
a)is the highlighted point always true?
b)if yes,could somebody please help me in proving the above..
c)if no,then is there any condition for which the tangent will touch one branch but cut the other branch
ii)I also seem to notice, that it is not possible to draw a tangent from any point on the corresponding asymptote of a branch of the hyperbola since the asymptote itself is parallel to the branch.
a)Is it possible to draw a tangent to the other branch of the hyperbola in the above situation?
i) a-b-c) A line can intersect a conic section at two points, at most, because the system of their equations is of second degree. Hence any tangent, which has a double point of contact with the conic, cannot have another intersection with the conic. NEVER.
ii) Yes, that's possible. Given a point outside the hyperbola, there are in general TWO lines through that point which are tangent to it. If the point lies on an asymptote, then one of those tangents is the asymptote itself, which can be regarded as a "tangent at infinity", but the other one is a "normal" tangent (unless, of course, that point is the intersection of the asymptotes).
Note, however, that you cannot assign an asymptote to a single branch of the hyperbola (as you seem to imply in your question): it is asymptotic to both branches at the same time.