A Short Treatise on the Conic Sections; In Which the Three Curves Are Derived from a General Description on a Plane, and the Most Useful Properties of Each Are Deduced from a Common Principle - Softcover

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated.1794 Excerpt: ... PROP. XXXVIII. The tangent at the vertex of any diameter of an hyperbola, which is terminated by the asymptotes, is equal to the conjugate diameter. LET PCG be any diameter of the hyperbola; Pl. mr. from P draw PD parallel to the asymptote FiG-64HC, meeting the conjugate hyperbola in D; and join CD. Through P draw the tangent VPH meeting the asymptotes in V and H. VP is equal to PH, Cor. 1. Prop. 35. and VP is to PH as PI is to HC; therefore PI is half of HC, but it is also half of PD, by the preceding proposition, therefore PD is equal to HC, and DC is equal and parallel to PH, and twice DC, or DK, is equal to 2/. Cor. 1. If DK be a conjugate diameter to PG, PG is also conjugate to DK. Join -D/and produce it till it meet. the asymptote in h. Uhen DC being equal and parallel to PH, or VP, VD is equal and parallel to PC, which is equal to Dh, it being the opposite side of the parallelogram PD /iC; therefore VDH touches the conjugate hyperbola in D, Cor. 2. Prop. 35. and PCG is a conjugate diameter to DCK. Cor. 2. Hence, if two tangents PV, DV be drawn through the vertices of any two conjugate diameters, they will meet in the asymptote; and the asymptotes are diagonals of the parallelogram which is formed by the four tangents. Cor. 3. If the hyperbolas be equilateral, the conjugate diameters will be equal: for the angle HCI will be a right angle, Cor. 2. Prop. 22. it will.-. theretherefore be in a semicircle, of which FHis the diameter and P the center; therefore CP is equal to PH, which is equal to CD. PROP. XXXIX. If through any point in an asymptote a right line be drawn cutting an hyperbola or opposite hyperbolas; the rectangle under the segments, between the asymptote and the hyperbola or hyperbolas, will be equal to the square of the semidiameter which ...