`z_1barz_2+barz_1z_2=2|z_1||z_2|cos(theta_1-theta_2)`

Let z_1=r_1(costheta_1+isintheta_1)a n dz_2=r_2(costheta_2+isintheta_2) be two complex numbers. Then prove that |z_1+z_2|^2=r1 2+r2 2+2r_1r_2cos(theta_1-theta_2) or |z_1+z_2|^2=|z_1|^2+|z_2|^2+2|z_1||z_2|^()_cos(theta_1-theta_2) |z_1-z_2|^2=r1 2+r2 2-2r_1r_2cos(theta_1-theta_2) or |z_1-z_2|^2=|z_1|^2+|z_2|^2-2|z_1||z_2|^()_cos(theta_1-theta_2)

Which of the following is (are) correct? (A) bar(z_1-z_2)-a(barz_1-barz_2)=0 (B) bar(z_1-z_2)+a(barz_1-barz_2)=0 (C) bar(z_1-z_2)+a(barz_1-barz_2)=-b (D) bar(z_1-z_2)+a(barz_1-barz_2)=-b

Let A(z_1), B(z_2) and C(z_3) forms an acute angled triangle in argand plane having origin as it’s orthocentre. Prove that z_1barz_2+bar(z)_1 z_2=z_2barz_3+barz_2z_3=z_3barz_1 +barz_3z_1

Modulus of a Complex Number & its properties If z;z_1;z_2inCC then (i)|z|=0hArrz=0 i.e. Re(z)=Im(z)=0 (ii)|z|=|barz|=|-z| (iii) -|z|leRe(z)le|z|;-|z|leIm(z)le|z| (iv) zbarz=|z|^2 (v)|z_1z_2|=|z_1||z_2| (vi)|(z_1)/(z_2)|=|z_1|/|z_2|; z_2!=0

If |z|=2 and locus of 5z-1 is the circle having radius a and z_1^2+z_2^2-2z_1z_2 cos theta=0, then |z_1|:|z_2|= (A) a (B) 2a (C) a/10 (D) none of these

Show that the equation of the circle in the complex plane with z_1 & z_2 as its diameter can be expressed as , 2z barz-(barz_1+barz_2)z-(barz_1+barz_2)barz+z_1barz_2+z_2barz_1=0

Prove that |(z_1, z_2)/(1-barz_1z_2)|lt1 if |z_1|lt1,|z_2|lt1

If |z|=2 and the locus of 5z-1 is the circle having radius 'a' and z_(1)^(2)+z_(2)^(2)-2z_(1)z_(2)cos theta=0 then |z_(1)|:|z_(2)|=

Let z_(1)=r_(1)(cos theta_(1)+i sin theta_(1)) and z_(2)=r_(2)(cos theta_(2)+i sin theta_(2)) be two complex numbers then prove the following

Let |(barz_1-2barz_2)/(2-z_1barz_2)|=1 and |z_2|!=1 where z_1 and z_2 are complex numbers show that |z_1|=2