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  4. Show that the area of the triangle on the argand diagram formed by the complex number z, iz and z + iz is (1/2)|z|2

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Q. Show that the area of the triangle on the argand diagram formed by the complex number $z, iz$ and $z + iz$ is $\frac{1}{2}|z|^2$

IIT JEEIIT JEE 1986Complex Numbers and Quadratic Equations

Solution:

We have, $i z=z e^{i \pi /2}$. This implies that $i z$ is the vector obtained by rotating vector $z$ in anti-clockwise direction through $90^{\circ}$. Therefore, $O A \perp A B$. So,
image
Area of $ \Delta O A B =\frac{1}{2} O A \times O B $
$=\frac{1}{2}|z||i z|=\frac{1}{2}|z|^{2} $