In case of the parabola this becomes—
The line joining any point on a parabola to the focus and the diameter through the point, are equally inclined to the tangent and normal at that point.
From the definition of a focus it follows that—
The segment of a tangent between the directrix and the point of contact is seen from the focus belonging to the directrix under a right angle, because the lines joining the focus to the ends of this segment are conjugate with regard to the conic, and therefore perpendicular.
With equal ease the following theorem is proved:
The two lines which join the points of contact of two tangents each to one focus, but not both to the same, are seen from the intersection of the tangents under equal angles.
§ 86. Other focal properties of a conic are obtained by the following considerations:
| Fig. 35. |
Let F (fig. 35) be a focus to a conic, f the corresponding directrix, A and B the points of contact of two tangents meeting at T, and P the point where the line AB cuts the directrix. Then TF will be the polar of P (because polars of F and T meet at P). Hence TF and PF are conjugate lines through a focus, and therefore perpendicular. They are further harmonic conjugates with regard to FA and FB (§§ 64 and 13), so that they bisect the angles formed by these lines. This by the way proves—
The segments between the point of intersection of two tangents to a conic and their points of contact are seen from a focus under equal angles.