Question

If \[{{z}_{1}},{{z}_{2}},{{z}_{3}},{{z}_{4}}\]are the affixes of four points in the Argand plane and \[z\] is the affix of a point such that \[\left| z-{{z}_{1}} \right|=\left| z-{{z}_{2}} \right|=\left| z-{{z}_{3}} \right|=\left| z-{{z}_{4}} \right|\], then \[{{z}_{1}},{{z}_{2}},{{z}_{3}},{{z}_{4}}\] are
A. concyclic
B. Vertices of a parallelogram
C. Vertices of a rhombus
D. In a straight line

Answer 1

Hint: In this question, we have to find the given complex numbers in the Argand plane. In order to find this, we need to consider the given condition of the points. So, that, we can able to define these complex numbers in the plane.

Formula Used: The complex number $(x,y)$ is represented by $x+iy$.
If $z=x+iy\in C$, then $x$ is called the real part and $y$ is called the imaginary part of $z$. These are represented by $\operatorname{Re}(z)$ and $\operatorname{Im}(z)$ respectively.
$z=x+iy$ be a complex number such that $\left| z \right|=r$ and $\theta $ be the amplitude of $z$. So, $\cos \theta =\dfrac{x}{r},\sin \theta =\dfrac{b}{r}$
And we can write the magnitude as
 $\begin{align}
  & \left| z \right|=\left| x+iy \right| \\
 & \Rightarrow r=\sqrt{{{x}^{2}}+{{y}^{2}}} \\
\end{align}$
Thus, we can write
$z=x+iy=r\cos \theta +ir\sin \theta =r(\cos \theta +i\sin \theta )$
This is said to be the mod amplitude form or the polar form of $z$.
Where $\cos \theta +i\sin \theta $ is denoted by $cis\theta $ and the Euler’s formula is $\cos \theta +i\sin \theta ={{e}^{i\theta }}$

Complete step by step solution: Given that, \[{{z}_{1}},{{z}_{2}},{{z}_{3}},{{z}_{4}}\] are the four points in the Argand plane.
Their relation with the affix of a point $z$ is
\[\left| z-{{z}_{1}} \right|=\left| z-{{z}_{2}} \right|=\left| z-{{z}_{3}} \right|=\left| z-{{z}_{4}} \right|\]
From this we can observe that, all these four points are equidistance from the affix of a point $z$.
So, the complex number $z$ is either the centre of a circle or the point of intersection of diagonals of a square.
Thus, the given four points in the plane are either concyclic or the vertices of a square.


Option ‘A’ is correct

Note: Here we need to remember that, the given four points are equally a parted from the $z$, the affix of a point. So, with this we can say that either the points are concyclic or vertices of a square.