The points `A,B,C` represent the complex numbers `z_1,z_2,z_3` respectively on a complex plane & the angle `B & C` of the triangle `ABC` are each equal to `1/2 (pi-alpha)` . If `(z_2-z_3)^2=lambda(z_3-z_1)(z_1-z_2) (sin^2) alpha/2` then determine `lambda`.

The points A,B and C represent the complex numbers z_(1),z_(2),(1-i)z_(1)+iz_(2) respectively, on the complex plane (where, i=sqrt(-1) ). The /_\ABC , is

If |z|=2, the points representing the complex numbers -1+5z will lie on

Complex numbers z_(1),z_(2)andz_(3) are the vertices A,B,C respectivelt of an isosceles right angled triangle with right angle at C. show that (z_(1)-z_(2))^(2)=2(z_1-z_(3))(z_(3)-z_(2)).

If the points represented by complex numbers z_(1)=a+ib, z_(2)=c+id " and " z_(1)-z_(2) are collinear, where i=sqrt(-1) , then

A,B and C are the points respectively the complex numbers z_(1),z_(2) and z_(3) respectivley, on the complex plane and the circumcentre of /_\ABC lies at the origin. If the altitude of the triangle through the vertex. A meets the circumcircle again at P, prove that P represents the complex number (-(z_(2)z_(3))/(z_(1))) .

Complex numbers z_(1),z_(2),z_(3) are the vertices of A,B,C respectively of an equilteral triangle. Show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

For complex number z =3 -2i,z + bar z = 2i Im(z) .

If z ne 1 and (z^(2))/(z-1) is real, the point represented by the complex numbers z lies

If the complex numbers z_(1) and z_(2) and the origin form an isosceles triangle with vertical angle (2pi)/(3) ,then show that z_(1)^(2)+z_(2)^(2)+z_(1)z_(2)=0.

The centre of circle represented by |z + 1| = 2 |z - 1| in the complex plane is