Question 1210364
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Let z1 and z2 be two complex numbers such that |z1| = 5 and z1/z2 + z2/z1 = 0. Find |z1 - z2|^2.
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        I will give here simple,  short,  clear and transparent solution.



<pre>
Let  complex number  z  be  z = z1//z2.   

Then  z2/z1  is  1/z,  and the given equation  z1/z2 + z2/z1 = 0  means

    z + {{{1/z}}} = 0,

    z = {{{-1/z}}}

    z^2 = -1

    z = {{{sqrt(-1)}}},  

    z = i  or  z = -i.  


If z = i,  then  z2 = i*z1,  and  

   |z1 - z2|^2 = |z1 - i*z1)^2 = |z1|^2*|1-i|^2 = 5^2*((1-i)*(1+i)) = 25*(1-i^2) = 25*(1-(-1)) = 25*(1+1) = 25*2 = 50.


If z = -i,  then  z2 = -i*z1,  and  

   |z1 - z2|^2 = |z1 + i*z1)^2 = |z1|^2*|1+i|^2 = 5^2*((1+i)*(1-i)) = 25*(1-i^2) = 25*(1-(-1)) = 25*(1+1) = 25*2 = 50.


<U>ANSWER</U>.  Under given conditions,  |z1-z2|^2 = 50.
</pre>

Solved.