In this video I go over further into the Unified the Theorem for Conics and this time prove it is applicable for hyperbolas. Recall that the theorem states that a conic is formed when the ratio of the distance to the focus divided by the distance to the directrix is a constant e, called the eccentricity. In my earlier videos I proved the case for parabolas (e = 1) and ellipses (e is less than 1), and in this video I prove the case for when e is greater than 1, i.e. for hyperbolas. But to save time I carry off from the proof of ellipses in which I first derived a formula for the conic for the general case of e is not equal to 1. The resulting formula in Cartesian or Rectangular coordinates looks very much like that for ellipses and hyperbolas, but the different values of e decides which conic is described. For the case that e is greater than 1, I show that indeed the unified theorem describes a hyperbola, and in fact it is a shifted horizontally shifted horizontal hyperbola. And just like for the ellipses case, the focus defined by the unified theorem is the exact same as that defined by the conventional theorem, albeit having two foci.
This is a very important video to understand how different theorems can describe the same curve but for better or worse across different coordinate systems. I will be going over some examples in later videos to better illustrate this theorem and its very simple formulation in polar coordinates so stay tuned!
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Conic Sections in Polar Coordinates: Theorem: Hyperbola Proof
In my previous videos I covered the proof of the Unified Theorem for Conic Sections in regards to the Parabola and the Ellipse, and in this video I will prove it is indeed applicable for hyperbolas.
Recall the conventional definition of a hyperbola and its formulation in standard Cartesian or Rectangular Coordinates:
Definition: A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F1 and F2 (the foci) is a constant.
Recall the Equation of a Horizontal Hyperbola in Standard Form:
Now let's recap on the Unified Theorem for Conic Sections which I covered in my earlier videos.
THEOREM:
Let F be a fixed point (called the focus) and L be a fixed line (called the directrix) in a plane.
Let e be a fixed positive number (called the eccentricity).
The set of all points P in the plane such that:
(that is, the ratio of the distance from F to the distance from L is the constant e)
is a conic section.
The conic is:
a) An ellipse if e < 1.
b) A parabola if e = 1.
c) A hyperbola if e > 1.
MES Note:
-- The eccentricity e is always positive since it is just the ratio of the distances.
The Theorem is illustrated below:
Also recall that the motivation behind the above theorem is that it can be written in a simple formula when using Polar Coordinates and setting the Focus at the Origin or Pole.
THEOREM in Terms of a Simple Polar Equation:
A polar equation of the form:
represents a conic section with eccentricity e and the Focus at the origin.
The conic is an ellipse if e < 1, a parabola if e = 1, or a hyperbola if e > 1.
The following figures illustrate the different polar equations for various conics:
Proof of Unified Conic Theorem for a Hyperbola: e > 1
Recall from my earlier video on the proof for ellipses that for the general case where e ≠ 1, we can convert the polar coordinates expression of the unified theorem into an expression that resembles that for ellipses and hyperbolas in Cartesian or Rectangular coordinates! #Amazing
Hyperbola: e > 1
If e > 1, then 1 - e2 < 0 and the above formula represents a hyperbola!
MES Note: We defined -h because the standard shifted conics formula includes the term (x - h)2. And unlike for ellipses, h > 0 in this case.
In fact, it is of the form of a Shifted Hyperbola with the center at (h , 0).
Recall that in the conventional Cartesian theorem for Hyperbolas, the foci are located a distance c from the center of the Hyperbola such that:
This confirms that the focus as defined in the above THEOREM means the same as the focus defined conventionally for Hyperbolas! as shown in my earlier video.
Also, we can see that the eccentricity can be written in terms of c and a.
MES Note #1: I made sure to take the result from the square root operation to obtain only positive values. WHAT WOULD HAPPEN IF WE TOOK THE NEGATIVE VALUES?? OR CONSIDERED BOTH?? #VeryVeryInteresting
MES Note #2: We can also find the above eccentricity formula as follows.
I will go over some examples in later videos to better illustrate this theorem and its polar coordinates form so stay tuned!!