All the proofs I've so far seen of the equation of the directrix of an ellipse based on its eccentricity (and also those which aren't) assume that the directrices are perpendicular to the focal axis. By those proofs I mean the ones using the product definition of the ellipse, the one that connects the distances from an arbitrary point on the ellipse to an arbitrary focus and the distances from that same point to a line, the driectrix.
I understand how c=ea and how, for a "horizontal lying" ellipse, x=(+/-)(a/e), but all of that given the directrices are parallel to the normal axis (i.e. minor axis/segment).
I tried deriving a general equation of an ellipse from the locus fact that the ratio of the distances from: a point on it to a focus, to a point on it and a line, is constant (e); all this with an arbitrary line L: px+qy+r=0, but ended up with a hot mess.
I was used to the eq of an ellipse being defined with the locus of constant foci distance equation and now this alternative definition comes up and I got confused. How do people know that the equation of the directrices are not skewed lines of the form y=mx+c? is it trivial/by definition that directrices are always at right angles to the ellipses' axes?