. Differential and integral calculus, an introductory course for colleges and engineering schools. and 7 are reciprocals one of the other. 8. p = asin|0. 9. The Cardioid: p = 2 a(l — cos0). Show that the length of a chord through the pole is constant. Showthat = 10, and that thereforetangents at the extremities of achord through the pole are per-pendicular to each other. Findthe points of maximum andminimum ordinates. Determinea simple geometrical construc-tion for this curve. 10. The Limacon of Pascal: p = b — a cos 6. There are four cases to becarefully considered, b <a,b = a, a<b <
. Differential and integral calculus, an introductory course for colleges and engineering schools. and 7 are reciprocals one of the other. 8. p = asin|0. 9. The Cardioid: p = 2 a(l — cos0). Show that the length of a chord through the pole is constant. Showthat = 10, and that thereforetangents at the extremities of achord through the pole are per-pendicular to each other. Findthe points of maximum andminimum ordinates. Determinea simple geometrical construc-tion for this curve. 10. The Limacon of Pascal: p = b — a cos 6. There are four cases to becarefully considered, b <a,b = a, a<b <2a,b = 2a. For each case draw two concentric circles of radii a and b and with thecenter at the pole, and thus get a simple geometric construction of thecurve. Show that the chord through the pole is of constant length. Find thepoints of contact of the double tangents when they exist. ii 2m 11. p = 1 — cos e 12. p = a sinf 0. The curve has eight double points and the pole is a multiple point. 13. p — a sin 10. The curve has six double points and the pole is a multiple 99 CURVES GIVEN BY POLAR EQUATIONS 139
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