Question

Find the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz .

• A. $\frac{1}{2}|z{|}^2$.
• B. $|z{|}^2$.
• C. $\frac{1}{3}|z{|}^2$.
• D. $\frac{1}{4}|z{|}^2$.

Answer 1

Answer: (A)

$\frac{1}{2}|z{|}^2$.

$\because i=cos\pi /2+isin\pi /2$
$\therefore i=e^{i\pi /2}$
$\therefore iz=z{e}^{i\pi /2}$
$\therefore$ Vector iz is obtained by rotating z in anticlockwise direction through 90${}^o$.
$\therefore |iz|=|i||z|=1.|z|=|z|=BC$
and $|z|=AB$
$\therefore AB=BC$, the triangle is an isosceles right angled triangle.
$\therefore Area=\frac{1}{2}\cdot AB\cdot BC$
$=\frac{1}{2}.|z|.|z|=\frac{1}{2}|z{|}^2$
Ans: A