. Plane and solid analytic geometry; an elementary textbook. Fig. 82. The sides of the parallelogram are evidently 2 af and 2 C drop a perpendicular CM on AB. Its length isthe distance from the origin to the line b2xxx + a2yxy = a2b2, or (by [IT]) car- a2p V6 V + a%2But the area of the parallelogram = 2 CM x ^5 = 2 (7i!f x P2KV 2 a262 = 4aJ. 2V&V + ^!2ab 152 ANALYTIC GEOMETRY [Ch. XI, § 82 Let the student give the proof for the 8. In the hyperbola the parallelogram formed by thetangents at the ends of conjugate diameters has its verticeson the asymptotes. 9. In the hyperbo
. Plane and solid analytic geometry; an elementary textbook. Fig. 82. The sides of the parallelogram are evidently 2 af and 2 C drop a perpendicular CM on AB. Its length isthe distance from the origin to the line b2xxx + a2yxy = a2b2, or (by [IT]) car- a2p V6 V + a%2But the area of the parallelogram = 2 CM x ^5 = 2 (7i!f x P2KV 2 a262 = 4aJ. 2V&V + ^!2ab 152 ANALYTIC GEOMETRY [Ch. XI, § 82 Let the student give the proof for the 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. Show that the angle between two conjugate diameters. i ab IS Sin —777- 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. Oh. XL § 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 that the points on either the major or minorauxiliar
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