If we next draw through A and B lines parallel to TF, then the points A1, B1 where these cut the directrix will be harmonic conjugates with regard to P and the point where FT cuts the directrix. The lines FT and FP bisect therefore also the angles between FA1 and FB1. From this it follows easily that the triangles FAA1 and FBB1 are equiangular, and therefore similar, so that FA : AA1 = FB : BB1.
The triangles AA1A2 and BB1B2 formed by drawing perpendiculars from A and B to the directrix are also similar, so that AA1 : AA2 = = BB1 : BB2. This, combined with the above proportion, gives FA : AA2 = FB : BB2. Hence the theorem:
The ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant.
To determine this ratio we consider its value for a vertex on the principal axis. In an ellipse the focus lies between the two vertices on this axis, hence the focus is nearer to a vertex than to the corresponding directrix. Similarly, in an hyperbola a vertex is nearer to the directrix than to the focus. In a parabola the vertex lies halfway between directrix and focus.
It follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the hyperbola it is greater than unity.
It is here the same which focus we take, because the two foci lie symmetrical to the axis of the conic. If now P is any point on the conic having the distances r1 and r2 from the foci and the distances d1 and d2 from the corresponding directrices, then r1/d1 = r2/d2 = e, where e is constant. Hence also (r1 ± r2) / (d1 ± d2) = e.
In the ellipse, which lies between the directrices, d1 + d2 is constant, therefore also r1 + r2. In the hyperbola on the other hand d1 − d2 is constant, equal to the distance between the directrices, therefore in this case r1 − r2 is constant.
If we call the distances of a point on a conic from the focus its focal distances we have the theorem:
In an ellipse the sum of the focal distances is constant; and in an hyperbola the difference of the focal distances is constant.
This constant sum or difference equals in both cases the length of the principal axis.