Hyperbola: A case of an ellipse? Can i treat a hyperbola as a special case of ellipse.
Like substituting $b^2$ with $-b^2$. Would all things still work?
And also,  why is a parabola different from the family of (circle, ellipse, hyperbola)? 
Or am I not looking at it correctly?  Thanks! 
 A: If you agree to work in the (real or complex) projective plane, the ellipse, the hyperbola and the parabola are essentially the same thing: consider, for simplicity, the ellipse $x^2 + y^2 = 1$ (a circle). Projectivize it, by replacing $x$ and $y$ by $\frac x z$ and $\frac y z$, with $z \ne 0$. You will get $x^2 + y^2 - z^2 = 0$. Good.
Think now of the hyperbola $x^2 - y^2 = 1$. Projectivizing it like above, you get $z^2 + y^2 = x^2$, so by a change the coordinates $(X,Y,Z) = (z,y,x)$ this becomes $X^2 + Y^2 - Z^2 = 0$, precisely like the ellipse above.
Now, let's take a look at the parabola $y = x^2$. Its projectivized equation is $zy = x^2$. Consider the change of variables $X = x, \ Y = \frac {y-z} 2, \ Z = \frac {y+z} 2$. Then the equation becomes $(Y+Z) (Z-Y) = X^2$, which again is just $X^2 + Y^2 - Z^2 = 0$, again like the ellipse.
If you don't want to work in the projective plane, but are at least willing to switch to complex coordinates, then the ellipse $x^2 + y^2 = 1$ is readily transformed into $X^2 - Y^2 = 1$ by the change of variables $x = X, \ y = \Bbb i Y$. You can't do this with the parabola, though. In the non-projective world, the parabola belongs to a different species that the ellipse and the hyperbola.
If you want to work neither with projective coordinates, nor with complex ones, then the three curves that you mention are completely different and there is no way to reconcile them.
The arguments illustrated above can be made to work on ellipses, hyperbolae and parabolae given by arbitrary equations, not just by the simple ones discussed, but I feel that for the purpouse of illustrating the underlying idea the chosen ones are just enough.
A: There is a unifying definition of the conics, based on a directrix line and a focus point, expressing that the ratio of the distance to the directrix andto the focus is a constant.
https://en.wikipedia.org/wiki/Conic_section#Eccentricity.2C_focus_and_directrix
When you increase the eccentricity, the conic which is first an ellipse starts growing and its center moves away from the directrix; at some point it goes to infinity, turning the ellipse to a parabola; then the center comes back from infinity on the other side while the curve changes to a hyperbola.
This is reflected in the equation of the conic in polar coordinates
$$\rho=\frac{l}{1+e\cos\theta}.$$
For $e<1$, the denominator doesn't cancel and this yields a closed curve, i.e. an ellipse. For $e>1$ there are two roots, corresponding to the two asymptotes of an hyperbola. And of course in between a single root yields the parabola.
In this representation, the three types of curves form a continuum. On the opposite, if you look at the centered conics, the parabola looks different as it "center" would be at infinity.
