Question

The area of the triangle whose vertices are complex numbers z, iz, z + iz in the Argand diagram is

• A. $2|z|^2$
• B. $ 1/2 |z|^2$
• C. $4|z|^2$
• D. $ |z|^2$

Answer 1

Answer: (B)

Vertices of triangle in complex form is $z, iz, z+iz$
In cartesian form vertices are
(x, y), (-y, x) and $(x-y, x+y)$
$\therefore $ Area of triangle $= \frac{1}{2}\begin{vmatrix}x&y&1\\ -y&x&1\\ x-y&x+y&1\end{vmatrix}$
$= \frac{1}{2}\left[x\left(x-x-y\right)-y\left(-y-x+y\right)+1\right]$
$\left(-yx-y^{2}-x^{2} + xy\right)$
$= \frac{1}{2}\left[-xy + xy-y^{2}-x^{2}\right]=\frac{1}{2} \left(x^{2}+y^{2}\right)$
($\because$ Area can not be negative)
$= \frac{1}{2}\left|z\right|^{2}\quad\quad\left(\because z = x+iy, \left|z\right|^{2} = x^{2}+y^{2}\right)$