PROP. If a point be assumed without or within an ellipse, and two right lines, parallel to two diameters, be drawn from it to cut or touch the ellipse ; then, as the rec tangle under the segments of the secant, or the square of the tangent, parallel to one of the diameters, is to the rectangle under the segments of the secant, or the square of the tangent, parallel to the other diameter, so is the square of the first dia meter to the square of the second diame ter. And the same thing is true of two transverse diameters of opposite hyperbo las, and any two lines, parallel to these, drawn through a point to cut the two curves.
For diameters of an ellipse, and of op posite hyperbolas, are secants that inter sect in the centre : and because they are bisected there, this proposition is mani fest from Pr. 10.
Def 15. Fig. 32. Let a point, as 0, be assumed in the plane of two opposite hy perbolas, and let the secant, 0 H K, be drawn through it parallel to a transverse diameter B A ; and the secants R 0 S, G 0 L, &c. parallel to any second diame ters, M N, P Q, &c. : in these diameters take the segments M N, P Q, &c. all bi sected in the centre, such that the squares of M N, P Q, &c. may severally be to the Square of the transverse diame ters A B, as the rectangles R 0 x 0 S, G 0 x 0 &c. contained by the seg ments of the secants parallel to the se cond diameters are to K 0 x 0 H, the rectangle under the segments of the secant parallel to the transverse diame ter then the magnitudes of the second diameters are the segments M N, P Q, &c.
Because the ratios of the rectangles KO x OH,SO x OR,G0 x OH, &c. are invariably the same wherever the point 0 is assumed, (10,) it is plain that the magnitudes of the second dia meters M N, P Q, &c. are also invaria bly the same wherever the point 0 is assumed.
And because the ratio of the rectan gles K 0 x 0 H to the square of the transverse diameter A B is the same as the ratio of the rectangle, contained by the segments of any secant drawn through 0, parallel to a transverse dia meter, to the square of the transverse diameter to which it is parallel, (19,) it is also manifest that the magnitudes of the second diameters are the same, from what ever transverse diameter they are de duced.
COr.1. And hence, taking the magni tudes of the transverse diameters as here defined, Prop. 19, may be enunciated for the hyperbola as generally as it is for the ellipse : that is, the rectangle under the segments of a secant, or the square of a tangent parallel to one dia meter (whether a transverse or a second diameter) of opposite hyperbolas, is to the rectangle under the segments of a secant, or the square of a tangent, paral lel to another diameter, as the square of the first diameter is to the square of the second diameter.
Car. 2. If two tangents be drawn to an ellipse, or a hyperbola, or opposite hyper bolas, from the same point, then these tangents are proportional to the diame ters, or semi-diameters, drawn parallel to the tangents.
For the squares of the tangents are proportional to the squares of the dia meters.
Cop. 3. If a right line be ordinately ap plied to a diameter of an ellipse, or to a transverse diameter of a hyperbola ; then as the square of the diameter is to the square of the conjugate diame ter, so is the rectangle contained by the abscisses of the diameter, between the vertices and ordinate, to the square of the ordinate.
For the double-ordinate is bisected by the diameter, and it is parallel to the con jugte diameter.
Fig. 33. If an ordinate be drawn to a second diameter of opposite hyperbolas : the square of this second diameter, is to the square of the conjugate diameter, as the sum of the squares of half the second diameter, and the part of it between the ordinate and the centre, is to the square of the ordinate.
Let A B and M N be conjugate diame ters of opposite hyperbolas, Il K an ordi nate to the second diameter M N, and draw 1n 1) S parallel to M N: then K D S is ordinately applied to A B (18); there fore