Question

If $z=x+iy$, then area of the triangle whose vertices are points $z,iz,z+iz$ is

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

Answer 1

The correct option is A

$\frac{1}{2}{\left|z\right|}^2$

Explanation for the correct option.

Step 1. Find the vertices of the triangle.

It is given that $z=x+iy$ is a complex number and its modulus is $\left|z\right|\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}$

Now $iz$ is defined as:

$\begin{array}{rcl}iz& =& i(x+iy)\\ & =& ix+i^{2}y\\ & =& ix+\left(-1\right)y\left[i^2=-1\right]\\ & =& -y+ix\end{array}$

And the complex number $z+iz$ is defined as:

$\begin{array}{rcl}z+iz& =& x+iy+\left(-y+ix\right)\\ & =& \left(x-y\right)+i\left(x+y\right)\end{array}$

As $z$, $iz$, and $z+iz$ are the vertices of a triangle, so the coordinates of the vertices are: $\left(x,y\right),\left(-y,x\right),\left(\left(x-y\right),\left(x+y\right)\right)$.

Step 2. Find the area of the triangle.

The area of the triangle having vertices as $\left(x,y\right),\left(-y,x\right),\left(\left(x-y\right),\left(x+y\right)\right)$ is given as:

$\begin{array}{rcl}\mathrm{Area}& =& \frac{1}{2}\left|\begin{array}{ccc}x& y& 1\\ -y& x& 1\\ x-y& x+y& 1\end{array}\right|\\ & =& \frac{1}{2}\left|\left[x\left(x-\left(x+y\right)\right)-y\left(-y-\left(x-y\right)\right)+1\left(-y\left(x+y\right)-x\left(x-y\right)\right)\right]\right|\\ & =& \frac{1}{2}\left|\left[x\left(x-x-y\right)-y(-y-x+y)+\left(-xy-y^2-x^2+xy\right)\right]\right|\\ & =& \frac{1}{2}\left|-xy+xy-y^2-x^2\right|\\ & =& \frac{1}{2}\left|-\left(x^2+y^2\right)\right|\\ & =& \frac{1}{2}\left(x^2+y^2\right)\\ & =& \frac{1}{2}{\left(\sqrt{x^2+y^2}\right)}^2\\ & =& \frac{1}{2}{\left|z\right|}^2\left[\left|z\right|\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}\right]\end{array}$

So, the area of the triangle whose vertices are points $z,iz,z+iz$ is $\frac{1}{2}{\left|z\right|}^2$.

Hence, the correct option is A.