Let `bar b z+b bar z=c, b ne 0`, be a line in the complex plane, where`bar b` is the complex conjugate of b. If a point `z_1` is the reflection of a point `z_2` through the line, then show that : `c= bar z_1 b+z_2 bar b`.

Show that arg. bar z =2 pi- arg. z .

Prove that : z= bar(z) iff z is real.

For a complex number z=x+ iy find the modulus of zbar(z) where bar(z) is the conjugate of z

Solve that equation z^2+|z|=0 , where z is a complex number.

If B is the mid point of bar (AC) and C is the mid point of bar(BD) where A,B,C,D lie on a straight line ,say why AB=CD ?

In complex plane, the equation |z+ bar z|=|z- bar z| represents :

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

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))) .

The points z_(1),z_(2),z_(3),z_(4) in the complex plane are the vertices of a parallelogram taken in order if and only if

Let a ,b and c be any three nonzero complex number. If |z|=1 and' z ' satisfies the equation a z^2+b z+c=0, prove that a .bar a = c .bar c and |a||b|= sqrt(a c( bar b )^2)