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How is $\frac{PF}{PD} = e = \frac{C}{A}$ ?

where e is eccentricity. What is the answer NOT using analytic geometry? (Using trigonometry)

P stands for any point on the ellipse. $F$ stands for one of the foci. $e$ stands for eccentricity. $D$ is a point on the directrix of the ellipse. 'C' is the distance from the center to the focus of the ellipse 'A' is the distance from the center to a vertex.

This is referring to an ellipse/hyperbola/parabola and their conic sections. The problem is not the proof for how $PF/PD = e$, or how $C/A = e$, but how the two equate to each other. (The letters stem from the points/foci of an ellipse of a cone and its directrix).

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Please provide enough context for us to understand what on earth you are talking about. –  Chris Eagle Apr 26 '12 at 19:23
To me it's impossible to undertand your question. Could you explain what PF, PD, C, A, etc. mean? –  Américo Tavares Apr 26 '12 at 19:25
It sounds like a conic section problem ... ? –  Neal Apr 26 '12 at 19:31
This is referring to an ellipse/hyperbola/parabola and their conic sections. The problem is not the proof for how PF/PD = e, or how C/A = e, but how the two equate to each other. (the letters stem from the points/foci of an ellipse of a cone and its directrix). –  Dona Apr 26 '12 at 19:33
What is C? What is A? –  Venge Apr 26 '12 at 20:27
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1 Answer

The figure below shows a conic section (specifically an ellipse) with focus $F$ and directrix $D$. The conic is the intersection of the blue cone and the purple "cutting plane". The figure also shows the Dandelin sphere associated with $F$ and $D$. (Interestingly, the Wikipedia article mentions the focus-directrix property "can be proved" with Dandelin spheres, but doesn't give that proof!) Let's call the green "horizontal" plane containing the circle where the sphere meets the cone the Dandelin plane; call that circle the Dandelin circle.

Now, given $P$ on the conic, let $Q$ be the corresponding point on the Dandelin circle; that is, let $Q$ be the point where the segment joining $P$ to the cone apex meets the Dandelin plane. Let $R$ be the foot of the perpendicular dropped from $P$ to the Dandelin plane, and let $S$ be the foot of the perpendicular dropped from $P$ to the directrix.

conic section

(Original image credit, with description of Dandelin spheres.)

Since segments $PF$ and $PQ$ are both tangent to the Dandelin sphere, we must have $PF=PQ$. (This, by the way, is the primary magic of the Dandelin sphere.)

We can massage the focus-directrix ratio for $P$ thusly:

$$\frac{PF}{PS} = \frac{PQ}{PS}=\frac{PR/(\sin\angle Q)}{PS}=\frac{PR}{PS}\frac{1}{\sin\angle Q}=\frac{\sin \angle S}{\sin \angle Q}$$

Clearly, $\angle S$ is constant as $P$ moves about the conic; it's the angle between the cutting plane and Dandelin plane. But $\angle Q$ is also constant: it's the ("exterior") angle that the surface (more precisely, a "generator" line) of the cone makes with the Dandelin plane. Therefore, the focus-directrix ratio is a constant.

To complete the answer to your question, all we have to do is prove that "(focal distance)-over-(semi-major axis)" gives the same trigonometric ratio.

For specificity, we'll assume the conic is an ellipse (that is $\angle S$ is smaller than $\angle Q$), leaving the reader to adjust the arguments as needed to deal with hyperbolas and parabolas.

Look at the figure "sideways", reducing all the elements to their intersections with the plane through the cone's axis, perpendicular to the directrix. The portion of the cone between apex and cutting plane appears as a triangle (which we'll label $\triangle ABC$); the Dandelin sphere appears as the incircle of that triangle. The ellipse's major axis is the segment $BC$ of that triangle and contains the focus $F$. Let's say that the incircle has center $O$ and radius $r$.

Projected figure

Segments $OB$ and $OC$ bisect the angles at $B$ and $C$ and serve as hypotenuses of right triangles with shared leg of length $r$. The other legs are therefore

$$BF = r \cot\frac{B}{2} \qquad CF = r \cot\frac{C}{2}$$

Note that, if $a$ is the length of the ellipse's semi-major axis, and $c$ the distance from its center to a focus, then $BF = a-c$ and $CF = a+c$, so that we can write

$$\begin{eqnarray} \frac{c}{a} &=& \frac{(CF-BF)/2}{(CF+BF)/2} =\frac{CF-BF}{CF+BF} =\frac{\cot\frac{C}{2}-\cot\frac{B}{2}}{\cot\frac{C}{2}+\cot\frac{B}{2}} \\ &=&\frac{\sin\frac{B}{2}\cos\frac{C}{2}-\cos\frac{B}{2}\sin\frac{C}{2}}{\sin\frac{B}{2}\cos\frac{C}{2}+\cos\frac{B}{2}\sin\frac{C}{2}} = \frac{\sin\left(\frac{B-C}{2}\right)}{\sin\left(\frac{B+C}{2}\right)} \end{eqnarray}$$

For the kicker, note that the cutting plane angle, $\angle S$, is preserved as the angle between $BC$ and "the horizontal"; likewise, the cone angle, $\angle Q$, is the angle between $AB$ (or $AC$) and the horizontal:

projected angles


$$\angle B = \angle Q + \angle S \qquad \text{and} \qquad \angle C = \angle Q - \angle S$$

(observe the importance of $\angle S$ being smaller than $\angle Q$ here) so that

$$\frac{B-C}{2} = \angle S \qquad \text{and} \qquad \frac{B+C}{2} = \angle Q$$

and we have

$$\frac{c}{a} = \frac{\sin\angle S}{\sin\angle Q}$$

which matches the focus-directrix ratio for points on the ellipse.

We happen to call this ratio the eccentricity of the conic, so this discussion reveals the geometric meaning of that number. I suspect that most students these days had no idea that there is such meaning. Kudos to the teacher who assigned this problem as homework.

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This explains how to solve for how PF/PD = e. But it doesn't prove C/A = e. I know you mentioned that now I simply have to prove that C/A = e is equal to the same trigonometric ratio. But I need a proof that actually describes out the process of going from PF/PD = e to C/A = e I realize I keep rewording myself but this isn't something I fully understand so I am trying to make it as easy as possible. –  Dona Apr 27 '12 at 0:04
@Dona: I don't want to give everything away in case this is homework. To prove $c/a = \sin S/\sin Q$, look at the figure "sideways", projecting into a plane perpendicular to the directrix. The cutting plane projects to a line; the cone --between apex (say, $A$) and cutting plane-- projects to a triangle (say, $\triangle ABC$); the Dandelin sphere projects to the incircle of that triangle. The ellipse's major axis projects to the "purple" segment ($BC$), and its center to the midpoint (say, $M$) of the segment. Use good ol' 2-dimensional trig to massage the ratio $MF/MB$ into $\sin S/\sin Q$. –  Blue Apr 27 '12 at 1:07
This was a homework assignment for my son last week, and since I couldn't help him then I figured I would try and understand it, and find out what the answer was. Even though it's past, I beleive that knowing the answer would maybe allow me to help him in the future (even though he is far beyond anything I ever learned). He already knew how to solve PF/PD=e but not how to get c/a=sinS/sinQ (I guess). Thank-you for continuing to help me. –  Dona Apr 27 '12 at 2:36
Thank-you so, so much, I could never have figured all this out on my own. I've been working on this for days, again; thank you VERY much. –  Dona Apr 27 '12 at 20:21
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