If `|(z-5i)/(z+5i)|=1` show that z is a real number

show that if |(z-3i)/(z+3i)|=1 then z is a real number.

If |z + i| = |z - i|, prove that z is real.

If z=x+iy and w=(1-iz)/(z-i), show that |w|=1o+_(z) is purely real.

If z=a+ib, |z|=1 and b ne 0, show that z can be represented as z=(c+i)/(c-i) where c is a real number.

Let z be a complex number having argument greater than 0 but less than 'pi/2' and satisfying equation (z-5i)(barz+5i)=25 then cot theta-10/z is equal to

Let z be a complex number satisfying |z-5i|<=1 such that amp(z) is minimum, then z is equal to

If (1+i)z=(1-i)phi z, then show that z=-iz

Fill in the blanks of the following . (i) For any two complex numbers z_(1), z_(2) and any real numbers a, b, |az_(1) - bz_(2)|^(2) + |bz_(1) + az_(2)|^(2) =* * * (ii) The value of sqrt(-25) xxsqrt(-9) is … (iii) The number (1 - i)^(3)/(1 -i^(3)) is equal to ... (iv) The sum of the series i + i^(2) + i^(3)+ * * * upto 1000 terms us ... (v) Multiplicative inverse of 1 + i is ... (vi) If z_(1) "and" z_(2) are complex numbers such that z_(1) + z_(2) is a real number, then z_(1) = * * * (vii) arg(z) + arg barz "where", (barz ne 0) is ... (viii) If |z + 4| le 3, "then the greatest and least values of" |z + 1| are ... and ... (ix) If |(z - 2)/(z + 2)| =(pi)/(6) , then the locus of z is ... (x) If |z| = 4 "and arg" (z) = (5pi)/(6), then z = * * *

Let z!=i be any complex number such that (z-i)/(z+i) is a purely imaginary number.Then z+(1)/(z) is