TITLE: Is there a parabola which is similar to a branch of hyperbola?
QUESTION [5 upvotes]: Parabola and a branch of hyperbola, visually looks similar. 
The only difference I find is that, when x tends to infinity, hyperbola approaches a straight line (asymptote). Whereas if I draw an arbitrary line the parabola will rush past that line and keep going away from it.
But still, is there any parabola which is exactly what a branch of hyperbola looks like?

REPLY [3 votes]: Actually parabola is a special case of hyperbola, where eccentricity tends to 1. 
Consider this equation of parabola. $$(x-2)^2+y^2=1•\dfrac{|x|^2}{1}$$ This is a parabola with focus at $(2,0)$ and equation of directrix $x=0$. This looks something like this.

Guess what this is

Looks similar to the above parabola, but it isn't. It's a hyperbola with same focus, same equation of directrix but with eccentricity 1.000001. Don't believe me see this zoomed out image. 

How far the second focus has reached, near to $-1\times 10^7$. As the eccentricity tends to 1, the second focus tends to $-\infty$ and the hyperbola looks more like a parabola.