In this video I go over further into Conics Sections in Polar Coordinates and this time prove that the Unified Theorem for Conics does in fact apply for parabolas.
The proof for parabolas is actually very simple because the Unified Theorem is derived very similarly to the conventional theorem for parabolas.
The Unified Theorem states that the ratio of the distance from the conic to a fixed point F (called the focus) divided by the distance from the conic to a fixed line L (called the directrix), |PF|/|PL| = e (called the eccentricity) is equal to 1 for parabolas; thus e = 1.
This means that |PF| and |PL| are “equidistant” or the same distances.
This result is in fact the exact same as the conventional definition of a parabola which states that a parabola is the set of points that are equidistant from the focus and the directrix.
Thus the ratio of the distance to the focus over the distance to the directrix must be 1 if they are equal.
While this proof is very simple, the proofs for the ellipse (e is less than 1) and hyperbolas (e is less than 1) are more complicated so make sure to stay tuned for those proofs!
