Plane and solid analytic geometry; an elementary textbook . the tangent at P2 can be shown to be parallel to PVKV This theorem appears also from the fact that the tan-gents are special cases of the system of parallel chords. 6. The tangent at the end of a diameter of the parabolais parallel to the system of chords which the diameter bisects. 7. The area of the parallelogram formed by tangents atthe ends of conjugate diameters of a central conic equals thearea of the rectangle on the principal axes. Oh. XI, § 82] DIAMETERS 151 Let ABED be the parallelogram formed by the tangentsat the ends of t
Plane and solid analytic geometry; an elementary textbook . the tangent at P2 can be shown to be parallel to PVKV This theorem appears also from the fact that the tan-gents are special cases of the system of parallel chords. 6. The tangent at the end of a diameter of the parabolais parallel to the system of chords which the diameter bisects. 7. The area of the parallelogram formed by tangents atthe ends of conjugate diameters of a central conic equals thearea of the rectangle on the principal axes. Oh. XI, § 82] DIAMETERS 151 Let ABED be the parallelogram formed by the tangentsat the ends of the conjugate diameters PXKX and P2K2. B^^ Y Ji sl—-—y^ \ ^\T^ \ T \ \ y^^^A D z^~~Ks Fig. 82. The sides of the parallelogram are evidently 2 a1 and 2 (7 drop a perpendicular (7M on AB. Its length isthe distance from the origin to the line b2xxx + a2^?/ = a2b2, or (by [17]) a%2 CM= ?Vb%2 + a4^2But the area of the parallelogram = 2 OMx AB=2 CMxP2K2, 2 a*b* -J 152 ANALYTIC GEOMETRY [Ch. XI, § Let the student give the proof for the hyperbola. r. 8. In the hyperbola the parallelogram formed by thetangents at the ends of conjugate diameters has its verticeson the asymptotes. 9. In the hyperbola, the line joining the ends of conju-gate diameters is parallel to one asymptote and is bisectedby the other. 10. Shoiv that the angle between two conjugate diametersi ab sin ] ab 11. Conjugate diameters of the rectangular hyperbola areequal. 12. The ellipse has a pair of equal conjugate diameterswhich coincide with the asymptotes of the hyperbola, whichhas the same axes as the ellipse. Ch. XI, § 82] DIAMETERS 153 PROBLEMS 1. Prove that conjugate diameters of an equilateral hyper-bola are equally inclined to the asymptotes. 2. Prove that any two perpendicular diameters of an equi-lateral hyperbola are equal. 3. Prove that the straight lines drawn from any point inan equilateral hyperbola to the extremities of any diameterare inclined at equal angles to the asymptotes. 4. Prove tha
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