Linear Algebra Solutions 59

Linear Algebra Solutions 59 - 11. Let a and b be distinct...

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11. Let a and b be distinct complex numbers and 0 < α < π . (i) When z 1 lies on the circular arc shown, it subtends a constant angle α . This angle is given by Arg( z 1 - a ) - Arg( z 1 - b ). However Arg µ z 1 - a z 1 - b = Arg( z 1 - a ) - Arg( z 1 - b ) + 2 = α + 2 kπ. It follows that k = 0, as 0 < α < π and - π < Arg θ π . Hence Arg µ z 1 - a z 1 - b = α. Similarly if z 2 lies on the circular arc shown, then Arg µ z 2 - a z 2 - b = - γ =
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