2 - Thus the equation |z |2 2 (az ) + |a|2 ) + |a|2 = 2 ,...

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Thus the equation | z | 2 - 2 ± ( az )+ | a | 2 )+ | a | 2 = ρ 2 , becomes ( x - u ) 2 +( y - v ) 2 = ρ 2 , which is the equation for a circle of radius ρ centered at ( u, v ), which in complex notation is the point a = u + iv . 5. We have | z | 2 =( ± z ) 2 +( ² z ) 2 ( ± z ) 2 , and taking square roots gives | z | | ± z | . Similarly for | z | | ² z | . We have | z + w | 2 =( z + w ) ( z + w ) = z z + z w + w z + w w = | z | 2 +2 ± ( z w )+ | w | 2 . Putting these equations together we have | z + w | 2 = | z | 2 +2 ± ( z w )+ | w | 2 | z | 2 +2 | ¯ zw | + | w | 2 = | z | 2 +2 | zw | + | w | 2 =( | z | + | w | ) 2 . 7. Recall that a circle in R 2 centered at ( a, b ) with radius r is given by the equation ( x - a ) 2 +( y - b ) 2 = r 2 . We will manipulate the equation
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This note was uploaded on 07/27/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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