If, on the other hand, we take a point P in the plane of a conic,
we get to each line a through P one conjugate line which joins P
to the pole of a. These pairs of conjugate lines through P form an
involution in the pencil at P. The focal rays of this involution are
the tangents drawn from P to the conic. This gives the theorem
reciprocal to the last, viz:—
A conic determines in every pencil in its plane an involution, corresponding lines being conjugate lines with regard to the conic.
If the point is without the conic the involution is hyperbolic, the tangents from the points being the focal rays.
If the point lies on the conic the involution is parabolic, the tangent at the point counting for coincident focal rays.
If the point is within the conic the involution is elliptic, having no focal rays.
It will further be seen that the involution determined by a conic on any line p is a section of the involution, which is determined by the conic at the pole P of p.
§ 83. Foci.—The centre of a pencil in which the conic determines a circular involution is called a “focus” of the conic.
In other words, a focus is such a point that every line through it is perpendicular to its conjugate line. The polar to a focus is called a directrix of the conic.
From the definition it follows that every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and every line joining two foci is an axis, for the perpendiculars to this line through the foci are conjugate to it. These conjugate lines pass through the pole of the line, the pole lies therefore at infinity, and the line is a diameter, hence by the last property an axis.
It follows that all foci lie on one axis, for no line joining a point in one axis to a point in the other can be an axis.
As the conic determines in the pencil which has its centre at a focus a circular involution, no tangents can be drawn from the focus to the conic. Hence each focus lies within a conic; and a directrix does not cut the conic.
Further properties are found by the following considerations:
§ 84. Through a point P one line p can be drawn, which is with regard to a given conic conjugate to a given line q, viz. that line which joins the point P to the pole of the line q. If the line q is made to describe a pencil about a point Q, then the line p will describe a pencil about P. These two pencils will be projective, for the line p passes through the pole of q, and whilst q describes the pencil Q, its pole describes a projective row, and this row is perspective to the pencil P.
We now take the point P on an axis of the conic, draw any line p through it, and from the pole of p draw a perpendicular q to p. Let q cut the axis in Q. Then, in the pencils of conjugate lines, which have their centres at P and Q, the lines p and q are conjugate lines at right angles to one another. Besides, to the axis as a ray in either pencil will correspond in the other the perpendicular to the axis (§ 72). The conic generated by the intersection of corresponding lines in the two pencils is therefore the circle on PQ as diameter, so that every line in P is perpendicular to its corresponding line in Q.
To every point P on an axis of a conic corresponds thus a point Q, such that conjugate lines through P and Q are perpendicular.
We shall show that these point-pairs P, Q form an involution. To do this let us move P along the axis, and with it the line p, keeping the latter parallel to itself. Then P describes a row, p a perspective pencil (of parallels), and the pole of p a projective row. At the same time the line q describes a pencil of parallels perpendicular to p, and perspective to the row formed by the pole of p. The point Q, therefore, where q cuts the axis, describes a row projective to the row of points P. The two points P and Q describe thus two projective rows on the axis; and not only does P as a point in the first row correspond to Q, but also Q as a point in the first corresponds to P. The two rows therefore form an involution. The centre of this involution, it is easily seen, is the centre of the conic.
A focus of this involution has the property that any two conjugate lines through it are perpendicular; hence, it is a focus to the conic.
Such involution exists on each axis. But only one of these can have foci, because all foci lie on the same axis. The involution on one of the axes is elliptic, and appears (§ 80) therefore as the section of two circular involutions in two pencils whose centres lie in the other axis. These centres are foci, hence the one axis contains two foci, the other axis none; or every central conic has two foci which lie on one axis equidistant from the centre.
The axis which contains the foci is called the principal axis; in case of an hyperbola it is the axis which cuts the curve, because the foci lie within the conic.
In case of the parabola there is but one axis. The involution on this axis has its centre at infinity. One focus is therefore at infinity, the one focus only is finite. A parabola has only one focus.
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| Fig. 34. |
§ 85. If through any point P (fig. 34) on a conic the tangent PT and the normal PN (i.e. the perpendicular to the tangent through the point of contact) be drawn, these will be conjugate lines with regard to the conic, and at right angles to each other. They will therefore cut the principal axis in two points, which are conjugate in the involution considered in § 84; hence they are harmonic conjugates with regard to the foci. If therefore the two foci F1 and F2 be joined to P, these lines will be harmonic with regard to the tangent and normal. As the latter are perpendicular, they will bisect the angles between the other pair. Hence—
The lines joining any point on a conic to the two foci are equally inclined to the tangent and normal at that point.
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:
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| 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.
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
