Also, I would like to know, can we apply the same for ellipse as well as parabola. If yes, what result we can obtain?
Just to get it out of the way: eccentricities are positive quantities, since they are defined as ratios of distances, which are positive quantities in themselves. Arturo has explained the hyperbolic case, so I shall confine myself to the other cases.
By definition, a parabola has unit eccentricity, so one cannot really speak of "parabolas of increasing eccentricity". In fact, parabolas are like circles, in that all parabolas are similar (differing only in scale).
For ellipses, the eccentricity $\varepsilon$ is constrained to lie in the interval $(0,1)$; one can thus study the behavior as $\varepsilon\to 0$ or $\varepsilon\to 1$.
To study the behavior of an ellipse as $\varepsilon\to 0$, consider the standard form for a central conic with center at the origin and axes lying on the coordinate axes:
where $a$ is the length of the principal axis. Taking the limit as $\varepsilon\to 0$ yields
and we have a circle of radius $a$. Geometrically, as the eccentricity shrinks, the two foci come closer and closer, up until they coincide with the center in the limit. (In this sense, a circle is a degenerate ellipse!)
As $\varepsilon\to 1$, to take the other extreme, if one fixes the position of one focus, the other focus will eventually shoot into the wild blue yonder (go to infinity), eventually leading to the parabola when $\varepsilon=1$.
To see this quantitatively, consider the general equation of a conic with one focus at the origin and principal axis coinciding with the horizontal axis:
$$(1-\varepsilon^2)x^2+y^2-2\varepsilon p x=p^2$$
where $p$ is the semilatus rectum. When $\varepsilon=1$, we have a parabola as expected. To see the behavior of an ellipse as $\varepsilon\to 1$, we treat the general equation as a generic central conic; we find that the coordinates of the "other focus" are
and it is apparent that the other focus does "shoot off to infinity" as $\varepsilon\to 1$.