Prove that `|z-z_(1)|^(2) + |z - z_(2)|^(2) = a` will represent a real circle on the Agrand plane if `2a ge |z_(1) -z_(2)|^(2)`

Prove that |Z-Z_1|^2+|Z-Z_2|^2=a will represent a real circle [with center (|Z_1+Z_2|^//2+) ] on the Argand plane if 2ageq|Z_1-Z_1|^2

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

The complex numbers z_(1), z_(2) and the origin are the vertices of an equilateral triangle in the Argand plane Prove that z_(1)^(2)+ z_(2)^(2)= z_(1).z_(2)

Let A(z_(1)) and B(z_(2)) are two distinct non-real complex numbers in the argand plane such that (z_(1))/(z_(2))+(barz_(1))/(z_(2))=2 . The value of |/_ABO| is

Let A(z_(1)),B(z_(2)),C(z_(3)) be the vertices of an equilateral triangle ABC in the Argand plane, then the number (z_(2)-z_(3))/(2z_(1)-z_(2)-z_(3)) , is

Prove that |z_(1) + z_(2)|^(2) + |z_(1)-z_(2)|^(2)=2|z_(1)|^(2) + 2|z_(2)|^(2)

If z_(1), z_(2) and z_(3) are the vertices of a triangle in the argand plane such that |z_(1)-z_(2)|=|z_(1)-z_(3)| , then |arg((2z_(1)-z_(2)-z_(3))/(z_(3)-z_(2)))| is

Prove that |z_1+z_2|^2=|z_1|^2+|z_2|^2, ifz_1//z_2 is purely imaginary.

Prove that |z_1+z_2|^2=|z_1|^2, ifz_1//z_2 is purely imaginary.

It is given the complex numbers z_(1) and z_(2) , |z_(1)| =2 and |z_(2)| =3 . If the included angle of their corresponding vectors is 60^(@) , then find value of |(z_(1) +z_(2))/(z_(1) -z_(2))|