The region of argand diagram defined by `|z-1|+|z+1|<=4 ` (1) interior of an ellipse (2) exterior of a circle (3) interior and boundary of an ellipse (4) none of these

Consider the region R in the Argand plane described by the complex number. Z satisfying the inequalities |Z-2| le |Z-4| , |Z-3| le |Z+3| , |Z-i| le |Z-3i| , |Z+i| le |Z+3i| Answer the followin questions : Minimum of |Z_(1)-Z_(2)| given that Z_(1) , Z_(2) are any two complex numbers lying in the region R is

Consider the region R in the Argand plane described by the complex number. Z satisfying the inequalities |Z-2| le |Z-4| , |Z-3| le |Z+3| , |Z-i| le |Z-3i| , |Z+i| le |Z+3i| Answer the followin questions : The maximum value of |Z| for any Z in R is

Let * be a binary operations on Z and is defined by , a * b = a + b + 1, a, bin Z . Find the identity element.

A relation R on the set of complex number is defined by z_1 R z_2 iff (z_1 - z_2)/(z_1+z_2) is real ,show that R is an equivalence relation.

Show that the area of the triangle on the Argrand Diagram formed by the complex numbers z,iz and z+iz" "is" (1)/(2)|z|^(2).

If z_1=a+ib and z_2=c+id are two complex numbers lying on the circle x^2+y^2=1 in the argand diagram and Re(z_1barz_2)=0 then the complex numbers omega_1=(a+ic) and omega_2=b+id are such that

P is a point on the argand diagram on the circle with OP as diameter two points Q and R are taken such that angle POQ = angle QOR = theta . If O is the origin and P, Q, R are are represented by complex z_1, z_2, z_3 respectively then show that z_2^2cos2 theta =z_1z_3 cos^2theta

If the vertices of an equilateral triangle be represented by the complex numbers z_(1),z_(2),z_(3) on the Argand diagram, then prove that, z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

Let z,z_(0) be two complex numbers. It is given that abs(z)=1 and the numbers z,z_(0),zbar_(0),1 and 0 are represented in an Argand diagram by the points P, P_(0) ,Q,A and the origin, respectively. Show that /_\POP_(0) and /_\AOQ are congruent. Hence, or otherwise, prove that abs(z-z_(0))=abs(zbar(z_(0))-1)=abs(zbar(z_(0))-1) .

Let z=x+i y be a complex number, where xa n dy are real numbers. Let Aa n dB be the sets defined by A={z :|z|lt=2}a n dB={z :(1-i)z+(1+i)bar z geq4} . Find the area of region AnnB