Question

The area of the triangle whose vertices are the points represented by the complex number $z,iz$ and $z+iz$ is

• A. $\frac{1}{4}|z{|}^2$
• B. $\frac{1}{8}|z{|}^2$
• C. $\frac{1}{2}|z{|}^2$
• D. $\frac{1}{2}|z|$

Answer 1

Answer: (C)

Area of the triangle is given by
$\Delta =\frac{1}{4}\left|\begin{array}{ccc}z& \bar{z}& 1\\ iz& -i\bar{z}& 1\\ z+iz& \bar{z}-i\bar{z}& 1\end{array}\right|$
Applying $R_3\to R_3-R_1-R_2$, we get
$\Delta =\frac{1}{4}\left|\begin{array}{ccc}z& \bar{z}& 1\\ iz& -i\bar{z}& 1\\ 0& 0& -1\end{array}\right|=\frac{1}{4}|(-1)\left|\begin{array}{cc}z& \bar{z}\\ iz& -i\bar{z}\end{array}\right|\mid$
$=\frac{1}{4}|(-1)(i)z\bar{z}|\left|\begin{array}{cc}1& 1\\ 1& -1\end{array}\right||=\frac{1}{4}z{|}^2(2)=\frac{1}{2}|z{|}^2$