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Real parameter given complex variable modulus 
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#1
Jan1711, 12:39 PM

P: 1

1. The problem statement, all variables and given/known data
Suppose z is complex number with z = 1 (also assume that z is not 1 + 0*i) Let z = a + b*i where a and b are real numbers. Find a real parameter, t, such that z = (ti)/(t+i), where i = sqrt(1) 2. Relevant equations z = sqrt(a^2+b^2) 3. The attempt at a solution Since we are given z = 1, this implies that a^2 + b^2 = (a+b*i)*(ab*i) = 1. Also, (ti)/(t+i) can be expressed as [1/(t^2+1)]*[(t^21)+(2*t)*i]. Despite my efforts, I end up at two deadends: I reduce an equation to a tautology like 0 = 0 or I reduce to an expression for t that involves i (thereby making t imaginary instead of real). Any help is greatly appreciated. 


#2
Jan1711, 04:09 PM

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P: 1,583

First off find c(t),d(t) such that:
[tex] c(t)+d(t)i=\frac{ti}{t+i} [/tex] Then use the fact that c(t)^{2}+d(t)^{2}=1 


#3
Jan1711, 05:11 PM

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P: 7,794

[tex]\frac{1}{t^2+1}\,\left[(t^21)+(2t)i\right][/tex] as [tex]\frac{(t^21)}{t^2+1}\frac{2t}{t^2+1}\,i\right][/tex] 


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