Question

Find |z| if area of a triangle formed by z,z+iz,iz is 50

Answer 1

Area of triangle formed by coordinate $(x_1,y_1)$,$(x_2,y_2)$ and $(x_3,y_3)$ is given by:
$A=\frac{1}{2}\begin{vmatrix}1&1&1\\x_1&x_2&x_3\\y_1&y_2&y_3\end{vmatrix}\\=\frac{1}{2}\left(\begin{vmatrix}x_2&x_3\\y_2&y_3\end{vmatrix}+\begin{vmatrix}x_3&x_1\\y_3&y_1\end{vmatrix}+\begin{vmatrix}x_1&x_2\\y_1&y_2\end{vmatrix}\right)$
Now, notice,
$\text{Im}(\bar{z_1}z_2)\\=\text{Im}{[(x_1-iy_1)(x_2+iy_2)]}\\=x_1y_2-x_2y_1\\=\begin{vmatrix}x_1&x_2\\y_1&y_2\end{vmatrix}$
Therefore,
$A=\frac{1}{2}[\text{Im}(\bar{z_1}z_2+\text{Im}(\bar{z_2}z_3)+\text{Im}(\bar{z_3}z_1))]\\=\frac{1}{2}[\text{Im}(\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1)]$
Remember this formula as it will be useful for finding the area of triangle in complex plane.
$\Rightarrow 50=\frac{1}{2}\text{Im}(\bar{z}z(1+i)+\bar{z}(1-i)iz-i\bar{z}z)\\\Rightarrow 100=\text{Im}(|z|^2(1+i)+|z|^2(1+i)+|z|^2(-i))\\\Rightarrow 100=\text{Im}(|z|^2(2+i))\\\Rightarrow 100=|z|^2\\\Rightarrow |z|=10$