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 March 27th, 2013, 07:53 AM #1 Senior Member   Joined: Feb 2013 Posts: 114 Thanks: 0 complex number - modulus problem If $|z_1+z_2|^2=|z_1-z_2|^2$ where $z_1$ and $z_2$ are non zero complex numbers, then which one is correct (a) Re$(\frac{z_1}{z_2})=0$ (b) Im $(\frac{z_1}{z_2})=0$ (c) $Re(z_1+z_2)=0$ Can we take $z_1= a+bi$ and $z_2= x+iy$ Now $|z_1+z_2|^2= |(a+x)+i(b+y)|^2 = (a+x)^2+(b+y)^2$.......(i) Also $|z_1-z_2|^2= |(a-x)+i(b-y)|^2 = (a-x)^2+(b-y)^2..$....(ii) Since (i) = (ii) After solving these two equation i get : 4ax +4by=0 .....(iii) What to do next ? could you please guide....
 March 27th, 2013, 11:15 AM #2 Senior Member   Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Re: complex number - modulus problem the correct answer is a. calculate z1/z2 and you get $\frac {a+bi}{x+yi} * \frac {x-yi}{x-yi} => \frac {ax+by-ayi+xbi }{x^2-y^2} Re( \frac {ax+by-ayi+xbi }{x^2-y^2})= \frac {ax+by}{x^2-y^2}$ you just proved that $ax+by=0$ so Re(z1/z2)=0

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