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  4. The area of the triangle on the complex plane formed by the points z, z+i z and i z is 128 . Then the value of |z| is

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Q. The area of the triangle on the complex plane formed by the points $z, z+i z$ and $i z$ is 128 . Then the value of $|z|$ is

KEAMKEAM 2021

Solution:

Let $z=x+i y$
$ z+i z=(x-y)+i(x+y) $
$ i z=-y+i x$
Area of triangle $=\frac{1}{2}\begin{vmatrix}x & y & 1 \\ x-y & x+y & 1 \\ -y & x & 1\end{vmatrix}=128$
$ x(x+y-x)-y(x-y+y)+1 $
$ \left(x^2-x y+x y+y^2\right)=256 $
$ x y-x y+x^2+y^2=256 $
$ x^2+y^2=256 $
$ \Rightarrow |z|^2=256 $
$ \therefore|z|=16$