If z=x+iy and |z|=1, show that the complex number `z_1=(z-1)/(z+1)` is purely imagine.

If z = x + iy and w=(1-iz)/(z-i) , show that : |w|=1 implies z is purely real.

If z_1,z_2 are comples numbers such that (2 z_1)/(3z_2) is purely imaginary number, then find |(z_1-z_2)/ (z_1+ z_2)| .

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

If |z|=1, prove that (z-1)/(z+1) (z ne -1) is purely imaginary number. What will you conclude if z=1?

If z_1 and z_2 are two complex number such that |(z_1-z_2)/(z_1+z_2)|=1 , Prove that iz_1/z_2=k where k is a real number Find the angle between the lines from the origin to the points z_1 + z_2 and z_1-z_2 in terms of k

If z=x+iy and |z+a|=3|z-a|, show that 2(x^2+y^2)=5ax-2a^2 .

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))) .

If w=alpha+ibeta, where beta!=0 and z!=1 , satisfies the condition that ((w- bar w z)/(1-z)) is a purely real, then the set of values of z is |z|=1,z!=2 (b) |z|=1a n dz!=1 (c) z=bar z (d) None of these

A complex number z is said to be unimodular if abs(z)=1 . Suppose z_(1) and z_(2) are complex numbers such that (z_(1)-2z_(2))/(2-z_(1)z_(2)^_) is unimodular and z_(2) is not unimodular. Then the point z_(1) lies on a

Let z_(1) and z_(2) be two distinct complex numbers and z=(1-t)z_(1)+tz_(2) , for some real number t with 0 lt t lt 1 and i=sqrt(-1) . If arg(w) denotes the principal argument of a non-zero complex number w, then a. abs(z-z_(1))+abs(z-z_(2))=abs(z_(1)-z_(2)) b. arg(z-z_(1))=arg(z-z_(2)) c. |{:(z-z_(1),bar(z)-bar(z)_(1)),(z_(2)-z_(1),bar(z)_(2)-bar(z)_(1)):}|=0 d. arg(z-z_(1))=arg(z_(2)-z_(1))