Question

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5. Let t be the inversive transformation defined by Determine the image of each of the following generalized circles under : (a) the extended line E U foo], where E is the line with equation y-x (b) the unit circle . 310 5: Inversive Geometry Problem 7 Let be the inversive transformation defined by 2-2i r(z) = 2. 2+2 Use the strategy to determine the image of each of the following generalized circles under : (a) the extended line aloo), where I is the line with equation x+y (b) the circle with equation x1)2y2 2; Here x + iy = z, as usual. 7. (a) To keep the calculations simple, we pick the points oo, 2i and 2 on the extended line, as shown below. Now 1, 1(2i)=0 1(00)= and 2-2i1 So the image of the extended line is the generalized circle that passes through the and 1-li. This is the circle of radius with centre 2i 0 (b) In this case, we pick the points -2,0 and 1 i on the circle, as shown below. Now 1(-2) = 00, 1(0) =-1 an 1 + i So the image of the circle is the gener- alized circle that passes through the points oo,-i and -1. This is the extended line with slope-1 through the point 0

Answer 1

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