four-point which is inscribed in both, and therefore one polar-triangle
common to both.
Theorem.—Two conics which intersect in four points have always one and only one common polar-triangle; and reciprocally,
Two conics which have four common tangents have always one and only one common polar-triangle.
Diameters and Axes of Conics
§ 69. Diameters.—The theorems about the harmonic properties of poles and polars contain, as special cases, a number of important metrical properties of conics. These are obtained if either the pole or the polar is moved to infinity,—it being remembered that the harmonic conjugate to a point at infinity, with regard to two points A, B, is the middle point of the segment AB. The most important properties are stated in the following theorems:—
The middle points of parallel chords of a conic lie in a line—viz. on the polar to the point at infinity on the parallel chords.
This line is called a diameter.
The polar of every point at infinity is a diameter.
The tangents at the end points of a diameter are parallel, and are parallel to the chords bisected by the diameter.
All diameters pass through a common point, the pole of the line at infinity.
All diameters of a parabola are parallel, the pole to the line at infinity being the point where the curve touches the line at infinity.
In case of the ellipse and hyperbola, the pole to the line at infinity is a finite point called the centre of the curve.
A centre of a conic bisects every chord through it.
The centre of an ellipse is within the curve, for the line at infinity does not cut the ellipse.
The centre of an hyperbola is without the curve, because the line at infinity cuts the curve. Hence also—
From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity. These are called Asymptotes (§ 59).
To construct a diameter of a conic, draw two parallel chords and join their middle points.
To find the centre of a conic, draw two diameters; their intersection will be the centre.
§ 70. Conjugate Diameters.—A polar-triangle with one vertex at the centre will have the opposite side at infinity. The other two sides pass through the centre, and are called conjugate diameters, each being the polar of the point at infinity on the other.
Of two conjugate diameters each bisects the chords parallel to the other, and if one cuts the curve, the tangents at its ends are parallel to the other diameter.
Further—
Every parallelogram inscribed in a conic has its sides parallel to two conjugate diameters; and
Every parallelogram circumscribed about a conic has as diagonals two conjugate diameters.
This will be seen by considering the parallelogram in the first case as an inscribed four-point, in the other as a circumscribed four-side, and determining in each case the corresponding polar-triangle. The first may also be enunciated thus—
The lines which join any point on an ellipse or an hyperbola to the ends of a diameter are parallel to two conjugate diameters.
§ 71. If every diameter is perpendicular to its conjugate the conic is a circle.
For the lines which join the ends of a diameter to any point on the curve include a right angle.
A conic which has more than one pair of conjugate diameters at right angles to each other is a circle.
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| Fig. 24. |
Let AA′ and BB′ (fig. 24) be one pair of conjugate diameters at right angles to each other, CC and DD′ a second pair. If we draw through the end point A of one diameter a chord AP parallel to DD′, and join P to A′, then PA and PA′ are, according to § 70, parallel to two conjugate diameters. But PA is parallel to DD′, hence PA′ is parallel to CC, and therefore PA and PA′ are perpendicular. If we further draw the tangents to the conic at A and A′, these will be perpendicular to AA′, they being parallel to the conjugate diameter BB′. We know thus five points on the conic, viz. the points A and A′ with their tangents, and the point P. Through these a circle may be drawn having AA′ as diameter; and as through five points one conic only can be drawn, this circle must coincide with the given conic.
§ 72. Axes.—Conjugate diameters perpendicular to each other are called axes, and the points where they cut the curve vertices of the conic.
In a circle every diameter is an axis, every point on it is a vertex; and any two lines at right angles to each other may be taken as a pair of axes of any circle which has its centre at their intersection.
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| Fig. 25. |
If we describe on a diameter AB of an ellipse or hyperbola a circle concentric to the conic, it will cut the latter in A and B (fig. 25). Each of the semicircles in which it is divided by AB will be partly within, partly without the curve, and must cut the latter therefore again in a point. The circle and the conic have thus four points A, B, C, D, and therefore one polar-triangle, in common (§ 68). Of this the centre is one vertex, for the line at infinity is the polar to this point, both with regard to the circle and the other conic. The other two sides are conjugate diameters of both, hence perpendicular to each other. This gives—
An ellipse as well as an hyperbola has one pair of axes.
This reasoning shows at the same time how to construct the axis of an ellipse or of an hyperbola.
A parabola has one axis, if we define an axis as a diameter perpendicular to the chords which it bisects. It is easily constructed. The line which bisects any two parallel chords is a diameter. Chords perpendicular to it will be bisected by a parallel diameter, and this is the axis.
§ 73. The first part of the right-hand theorem in § 64 may be stated thus: any two conjugate lines through a point P without a conic are harmonic conjugates with regard to the two tangents that may be drawn from P to the conic.
If we take instead of P the centre C of an hyperbola, then the conjugate lines become conjugate diameters, and the tangents asymptotes. Hence—
Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.
As the axes are conjugate diameters at right angles to one another, it follows (§ 23)—
The axes of an hyperbola bisect the angles between the asymptotes.
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| Fig. 26. |
Let O be the centre of the hyperbola (fig. 26), t any secant which cuts the hyperbola in C, D and the asymptotes in E, F, then the line OM which bisects the chord CD is a diameter conjugate to the diameter OK which is parallel to the secant t, so that OK and OM are harmonic with regard to the asymptotes. The point M therefore bisects EF. But by construction M bisects CD. It follows that DF = EC, and ED = CF; or
On any secant of an hyperbola the segments between the curve and the asymptotes are equal.
If the chord is changed into a tangent, this gives—
The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact.
The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. This is equivalent to three points and the tangents at two of them. This construction requires measurement.
§ 74. For the parabola, too, follow some metrical properties. A diameter PM (fig. 27) bisects every chord conjugate to it, and the pole P of such a chord BC lies on the diameter. But a diameter cuts the parabola once at infinity. Hence—
The segment PM which joins the middle point M of a chord of a parabola to the pole P of the chord is bisected by the parabola at A.
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| Fig. 27. |
§ 75. Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the
