Here is probably a very stupid question, but I couldn't find the answer myself. Also, I was unable of finding anywhere.

I bet all of you knows this obvious stuff that you are about to read, but I'll write anyway: Let $F_1$ and $F_2$ and $F$ to be arbitrarily non-coincident points. Let $L$ and $D$ be the directrix lines such that $F\notin L$ and $F\notin D$. The point $P$ belongs to the conic section. Notation $d(A, B)$ means the distance between geometrical objects $A$ and $B$ (points or lines).

$$ \mbox{Conics: } \begin{cases} \mbox{Ellipse: } && d(P, F_1) + d(P, F_2) = 2a \\ \mbox{Hyperbola: } && d(P, F_1) - d(P, F_2) = 2a \\ \mbox{Parabola: } && d(P, F) = d(P, L) \\ \end{cases} $$

For the appropriate values of $\epsilon$, we have each of those conics, with a single equivalent definition: $$d(P, F) = \epsilon\cdot d(P, D)$$.

Well, one can prove it by selecting point $P = (x, y)$, making the proper equations, with a few simplifications, on the first definition, and on the second, and show they are the same.

Here is my question: How can one prove equivalence between those two definitions very quickly? Efortlessly?

Also, if you are feeling generous, perhaps you can tell me: is there a way of proving this, using the whole time the $d(A, B)$ notation? How? =).

The little of what I got:

The proof for parabola is obvious. Then, basically, we have: $$ \mbox{Conics: } \begin{cases} \mbox{Ellipse: } && d(P, F_1) = 2a - d(P, F_2) = \epsilon\cdot d(P, D), && \epsilon < 1 \\ \mbox{Hyperbola: } && d(P, F_1) = 2a + d(P, F_2) = \epsilon\cdot d(P, D), && \epsilon > 1 \\ \end{cases} $$

Thus, we are basically proving that, for fixed point $F$, fixed line $D$, fixed constant $a>0$, there exists $\epsilon$ such that: $$ 2a \pm d(P, F) = \epsilon\cdot d(P, D), \quad\forall P\in\mathbb{R}^2 $$

Well.. that's it. I couldn't progress further. I'll edit if I make progresses....

  • $\begingroup$ Actually, I've never seen the second formula for conic sections. What would be appropriate values of $\epsilon$ for ellipses and hyperbolas? Furthermore, what are $F$ and $D$ (geometrically) for ellipses and hyperbolas? $\endgroup$ – A.P. Oct 8 '15 at 19:56
  • $\begingroup$ @A.P. The value $\epsilon$ is called eccentricity of the conic section. For $0\le\epsilon < 1$, it become an ellipse. For $\epsilon>1$ become an hyperbola. $F$ is the foci. $D$ is a directrix line. $\endgroup$ – Physicist137 Oct 8 '15 at 22:35
  • $\begingroup$ I didn't use the same variables you did, but I've proved this statement for ellipses and hyperbolas in this question and answer. $\endgroup$ – Noble Mushtak Jun 21 '16 at 0:54

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