Question

The area of the triangle on the complex plane formed by the complex numbers $z,iz$ and $z+iz$, is

• A. $|z|$
• B. $|\bar{z}{|}^2$
• C. $\frac{1}{2}|z{|}^2$
• D. $\frac{1}{2}|z+iz{|}^2$

Answer 1

Answer: (C)

Let $z=x+iy$. Then, $iz=-y+ix$. Therefore,
$z+iz=(x-y)+i(x+y)$
Required area of triangle
$=\text{ Area of triangle formed by }(x,y)(-y,x)\text{ and }(x-y,x+y)$
$=\frac{1}{2}|x(x-x-y)-y(x+y-y)+(x-y)(y-x)|$
$=\frac{1}{2}\left|2xy-2xy-x^2-y^2\right|$
$=\frac{1}{2}\left(x^2+y^2\right)=\frac{|z{|}^2}{2}$
Geometrically

Area of $△ABC-\frac{1}{2}$ area of square $O\Lambda CB$
$=\frac{1}{2}|z{|}^2$