Show that: `sum_(r=0)^(n-1)|z_1+alpha^r z_2|^2=n(|z_1|^2+|z_2|^2),w h e r e, alpha ; r=0,1,2,...,(n-1)` , are the nth roots of unity and `z_1, z_2` are any two complex numbers.

If z_1=5+12 i and |z_2|=4,t h e n

If |z_1|=15 and |z_2-3-4i|=5,t h e n

If 1,z_1 ​ ,z_2 ​ ,z_3 ​ ,...,z_(n−1) ​ are nth roots of unity, then show that (1−z_1 ​ )(1−z_2 ​ )...(1−z_(n−1) ​ )=n

Let z_1 and z_2 be the nth roots of unity which subtend a right angle at the origin then n must be the form

If |z_(1)|=|z_(2)|and argz_(1)+argz_(2)=0, then show that z_(1) and z_(2) are tow complex conjugate numbers. '

If |z_1|=|z_2|=1, then prove that |z_1+z_2| = |1/z_1+1/z_2∣

If z_r : r=1,2,3, ,50 are the roots of the equation sum_(r=0)^(50)z^r=0 , then find the value of sum_(r=0)^(50)1/(z_r-1)

If z_1 + z_2 + z_3 + z_4 = 0 where b_i in R such that the sum of no two values being zero and b_1 z_1+b_2z_2 +b_3z_3 + b_4z_4 = 0 where z_1,z_2,z_3,z_4 are arbitrary complex numbers such that no three of them are collinear, prove that the four complex numbers would be concyclic if |b_1b_2||z_1-z_2|^2=|b_3b_4||z_3-z_4|^2 .

Let |(( bar z _1)-2( bar z _2))//(2-z_1( bar z _2))|=1 and |z_2|!=1 ,where z_1 and z_2 are complex numbers. Show that |z_1|=2.

If the fourth roots of unity are z_1,z_2,z_3,z_4 then z_1^2+z_2^2+z_3^2+z_4^2 is equal to