If `z_1a n dz_2` are conjugate to each other then find `a r g(-z_1z_2)dot`

If z_1a n dz_2 are two complex numbers such that |z_1|=|z_2|a n d arg(z_1)+a r g(z_2)=pi , then show that z_1,=-( barz )_2dot

If z_1a n dz_2 are two complex numbers and c >0 , then prove that |z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot

If z_1a n dz_2 are two nonzero complex numbers such that = |z_1+z_2|=|z_1|+|z_2|, then a rgz_1-a r g z_2 is equal to -pi b. pi/2 c. 0 d. pi/2 e. pi

If z_1a n dz_2 are complex numbers and u=sqrt(z_1z_2) , then prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|

Let z_1a n dz_2 be complex numbers such that z_1!=z_2 and |z_1|=|z_2|dot If z_1 has positive real part and z_2 has negative imaginary part, then (z_1+z_2)/(z_1-z_2) may be (a)zero (b) real and positive (c) real and negative (d) purely imaginary

If z_1a n dz_2 are the complex roots of the equation (x-3)^3+1=0,t h e nz_1+z_2 equal to a. 1 b. 3 c. 5 d. 7

If z_(1) and z_(2) are 1 - i, -2 + 4i then find Im ((z_(1) z_(2))/(bar(z_(1)))) .

z_1a n dz_2 are two distinct points in an Argand plane. If a|z_1|=b|z_2|(w h e r ea ,b in R), then the point (a z_1//b z_2)+(b z_2//a z_1) is a point on the (a)line segment [-2, 2] of the real axis (b)line segment [-2, 2] of the imaginary axis (c)unit circle |z|=1 (d)the line with a rgz=tan^(-1)2

If z_(1)" and "z_(2) are two complex numbers such that |(z_(1)-z_(2))/(z_(1)+z_(2))|=1 then

z_1a n dz_2 lie on a circle with center at the origin. The point of intersection z_3 of the tangents at z_1a n dz_2 is given by a. 1/2(z_1+( z )_2) b. (2z_1z_2)/(z_1+z_2) c. 1/2(1/(z_1)+1/(z_2)) d. (z_1+z_2)/(( z )_1( z )_2)