Is there a shortcut to get the second directrix of an ellipse in polar form? The general form for conics in polar coordinates indicate only one directrix depending on the type of equation.



With ellipses, to my knowledge there are suppose to be two directrices.
I've tried to do an equation of an ellipse in Geogebra, where the original equation features a directrix of x = 4. I've tried to see if x = -4 is also a directrix. It shows a line tangent to the ellipse, which doesn't seem to be a directrix since it should be equidistant with the actual x = 4 directrix to the center of the ellipse.
I know you can probably just convert the equation to become in terms of rectangular coordinates and then find the second directrix through the conventional solution of adding and subtracting the directrix distance at the center on the coordinate along the major axis. But I was wondering if there is a shortcut, since most sources just consider one directrix for the ellipse.
 A: There's no need to convert coordinate systems to do the distance arithmetic. The polar equation encodes all the information you need.

Ignoring the parabola for now, consider a central conic with eccentricity $e$, major/transverse radius $a$, center-to-focus distance $c:=ae$, center-to-directrix distance $d:=a/e$. Then,
$$
\frac{\text{focus-to-other-directrix dist}}{\text{focus-to-directrix dist}}=\frac{c+d}{|c-d|}=\frac{e^2+1}{|e^2-1|} \quad=:\lambda$$
In you formulas, the focus-to-directrix distance is $p$, so that the focus-to-other-directrix distance is $\lambda p$.
If we drop the absolute value bars, the sign of $\lambda$ conveniently tells us whether the two directrices are on the same side of the focus (positive; when $e>1$, and the conic is a hyperbola) or on opposite sides of the focus (negative; $e<1$, ellipse).
Note that the ratio us undefined (arguably, infinite) when $e=1$, corresponding to the parabola. This is consistent with the notion that the "other" directrix is the "line at infinity" (which is simultaneuosly on the same and opposite sides of the focus as the other directrix).
