Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)

By Milgram R. (ed.)

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Prove that the mid-points of all chords of the ellipse parallel to AB lie on another diameter CD , and find the equation of CD . Prove that, for all m, AB 2 + CD 2 is constant. P is any point on the ellipse and the tangent at P meets the coordinate axes at Q, R . If P is the mid-point of QR, show that P lies on a diagonal of the rectangle which circumscribes the ellipse and has its sides parallel to the axes of coordinates. P, and P2 are two points on the ellipse such that ()has the value (), at P, and the value O2 at P2.

3z 2 - 6z + 10 = 0, Note that here, conventionally, we express the equation in terms of z, since z may be complex . (1 - i)" + 3(1 - i)2 - 6(1 - i) + 10 = (-2i)2 - 6i - 6 + 6i + 10 = -4 - 6 + 10 = O. Hence 1 - i is a root, and hence also is 1 + i, (z - 1 + i)(z - 1 - i) == Z2 - 2z + 2. Z4 + 3z 2 - 6z + 10 == (Z2 - 2z + 2)(Z2 + 2z + 5) => remaining roots are - 1 ± 2i. 4 The Argand diagram The French mathematician Argand realised that the coordinate system introduced by Descartes, in which two perpendicular axes are used as number lines Complex numbers 55 to describe the position of a point in their plane , provides a convenient way of representing complex numbers diagrammatically.

3 1 Sketch the curve r 2 = a 2 cos 26, where r ~ 0 and a > o. (a) Find the angle between the tangents to the curve at the pole. (b) Find the area of the region enclosed by one loop of the curve. /4. Find the area of C. State equations of the tangents to the curve r = 4 cos 26 at o. /4, and r = a, where a> O. Show that the area of the finite region lying within both curves is a 2 (4rr. J3)/12. Sketch the curve r = acos 36, where a > 0, showing clearly the tangents to the curve at the pole. Find the area of the finite region enclosed by one loop of the curve.

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