ft. |z- 1+3 i 2 |= 10 2 ) If (P, Q ) and (R ) are points, respectively representing the complex numbers (z, z e^ i 3 ) and (z (1+e^ i 3 ) ) in argand plane, then the area of the triangle (P Q R ), is

Step-by-step solution

Given that, ( aligned P Q & = |z e^ i / 3 -z |=|z| |e^ i / 3 -1 | & =|z| | 3 +i 3 -1 | & =|z| |-2 ^2 6 +2 i 6 6 | & =|z| |2 6 | | 6 -i 6 |=|z| |2 1 2 ||1| aligned ) (P Q=|z| ) Now, ( aligned & Q R= [z (1+e^ i / 3 )-2 e^ i / 3 ] & Q R=|z| aligned ) Similarly, (P R=|z| ) so, (P Q=Q R=P R ) ( P Q R ) is equilateral triangle with side length ( 1 ). Now, area of ( P Q R= 3 4 |z|^2 )