Download Advanced Mathematics 3 by C. W. Celia, A. T. F. Nice, K. F. Elliott PDF

By C. W. Celia, A. T. F. Nice, K. F. Elliott

ISBN-10: 0333348273

ISBN-13: 9780333348277

ISBN-10: 1349067113

ISBN-13: 9781349067114

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Example text

The centre of the circle is given by z = 4e/3 and its radius is tlcl. r P x Fig. 11 CP = iOP The equation [z - pi = klz - ql, where k ¥= I, represents a circle with radius kip - ql/II - k 2 1 and with its centre at the point given by z = (p - qk2)/(I - k 2 ) . This can be shown by the method above, or by considering the Complex numbers 43 transformation Z = z - q. This type oftransfonnation is considered in the next section. A circle defined in this way is known as a circle of Apollonius. Example 3 Find the cartesian equation of the circle [z - 31 = 21z - i].

2' Hence as z moves from A to B, w moves along the arc AIB I shown in Fig . 21. (c) On BC, let z =i+ t, where 0 :::; t :::; I. t - i I This gives w=--=--- => u=--- i + ( t I + (2 ' I + (2 - I v=--I + /2 Complex numbers 55 1 =---= 1 + t2 => => u2 + (v + t)2 = -v t· This is the equation of the circle in the w-plane with centre given by w = - i/2 and with radius t . As t decreases from 1 to 0, u decreases from t to 0 and v decreases from- t to -1. Hence in Fig. C! (d) On CO , let z = it, where 0 ~ t ~ 1.

21. (c) On BC, let z =i+ t, where 0 :::; t :::; I. t - i I This gives w=--=--- => u=--- i + ( t I + (2 ' I + (2 - I v=--I + /2 Complex numbers 55 1 =---= 1 + t2 => => u2 + (v + t)2 = -v t· This is the equation of the circle in the w-plane with centre given by w = - i/2 and with radius t . As t decreases from 1 to 0, u decreases from t to 0 and v decreases from- t to -1. Hence in Fig. C! (d) On CO , let z = it, where 0 ~ t ~ 1. Then w = l iCit) = -i/t. As t decreases from 1 to 0, w moves along the negative v-axis away from the origin.

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Advanced Mathematics 3 by C. W. Celia, A. T. F. Nice, K. F. Elliott


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