Let p  and q  be two complex numbers such that q  ≠  0.  If the roots of the equation x2+px+q2=0  have the same absolute value, then p/q  is

Let x1,x2  be the roots of x2+px+q2=0  and |x1|=|x2|=r.  Note that x1x2=q2≠0.  Now, p2q2=(x1+x2)2x1x2=x1x2+x2x1+2                                          =x1x2¯|x2|2+x1¯x2|x1|2+2=1r2(x1x2¯+x1¯x2)+2                                         =2r2Re(x1x¯2)+2⇒  p2q2  is a real number.Next, as −Re  (x1 x¯2) ≤ |x1  x¯2|  =  |x1  x2|=|x1||x2|=r2,we get ⇒p2q2≥−2r2r2+2=−2+2=0Therefore, p/q  is a real number.