If `z_1, z_2a n dz_3, z_4` are two pairs of conjugate complex numbers, prove that `a rg((z_1)/(z_4))+a rg((z_2)/(z_3))=0.`

Explanation

Given `{z}_{1} {z}_{2} {z}_{3}, {z}_{4}` are conjugate complete no ;



`therefore z_{2}=bar{z}_{1} z_{4}=bar{z}_{3} `


`arg (frac{z_{1}}{z_{4}})+arg (frac{z_{2}}{z_{3}})=arg (frac{z_{1}}{z_{4}})(frac{z_{2}}{z_{3}}) `


`=arg (frac{z_{1}}{z_{3}})(frac{bar{z}_{1}}{z_{3}}) `


`=arg (frac{z_{1} bar{z}_{1}}{z_{3} bar{z}_{3}})`


`=arg (frac{|z_{1}|^{2}}{.z_{3}|^{2}}) `


`=0(frac{|z_{1}|^{2}}{|z_{3}|^{2}} Rightarrow text { purely real })`