question_answer

Area of the triangle in the argand diagram formed by the complex numbers\[z,iz,z+iz,\] where\[z=x+iy,\]is:

A)  \[|z|\]                

B)                         \[|z{{|}^{2}}\]                 

C)         \[2|z{{|}^{2}}\]                               

D)         \[\frac{1}{2}|z{{|}^{2}}\]

E)         \[\frac{1}{2}|z|\]

Correct Answer: D

Solution :

\[iz=z{{e}^{i\pi /2}}\] This implies that 12 is the vector obtained by rotating vector z in anticlock wise direction through\[90{}^\circ \]. \[\therefore \]  \[OA\bot AB\] So area of \[\Delta OAB=\frac{1}{2}OA\times OB\]                                 \[=\frac{1}{2}|z||iz|=\frac{1}{2}|z{{|}^{2}}\]