Geometry of Complex Numbers

The complex numbers  $z_1,z_2$  and  $z_3$  satisfying  $\frac{z_1-z_3}{z_2-z_3}=\frac{\mathrm{}1-i\sqrt{3}}{2},$  are the vertices of a triangle which is

Check whether the given four points in a complex plane form a rectangle or not.

$z_1=1+5i$, $z_2=1+2i$, $z_3=-2+2i$, $z_4=-2+6i$ are the points.

The given four points in a complex plane form a rectangle.

$z_1=5+4i$, $z_2=5-4i$, $z_3=-2-4i$, $z_4=-2+4i$ are the points.

Check whether the given four points in a complex plane form a rectangle or not.

$z_1=2+2i$, $z_2=2-2i$, $z_3=-3-2i$, $z_4=-3+2i$ are the points.

Let the affix of $\left(1-2i\right)$ be $P$. Then $OP$ is rotated about origin $O$ through an angle of $180°$ in anti-clockwise direction. The new complex number is

Complex number $z=3+2i$

Plot the $iz$ on complex plane.

The figure in the complex plane given by $10z\overline{z}-3\left(z^2+{\overline{z}}^2\right)+4i\left(z^2-{\overline{z}}^2\right)=0$, is

Let $z=x+iy$ and $w=u+iv$ be complex numbers on the unit circle such that $z^2+w^2=1$. Then the number of ordered pairs $\left(z,w\right)$ is

If $z_1=2-3\mathrm{i}$ and $z_2=-1+i,$ then the locus of a point $P$ represented by $z=x+iy$ in the Argand plane satisfying the equation $\mathit{Arg}\left(\frac{z\mathit{-}{z}_{\mathit{1}}}{z\mathit{-}{z}_{\mathit{2}}}\right)\mathit{=}\frac{\pi }{\mathit{2}}$ is

The locus of $z$ represented by $Re\left(z^2-2z\right)=15$ is

If $w=\frac{z-3}{2z-i}$ and $\left|w\right| = 1,$ then the locus of $z$ is given by :

How many complex numbers $z,$ satisfies both the equations $\left|z+5\right|+\left|z-5\right|=10$ and $\left|z+1\right|=2$ ?

The locus of the complex numbers $z$ which satisfies ${\left|z+10i\right|}^2 – {\left|z –10i\right|}^2=20$, is (where $i=\sqrt{-1}$)

Consider a square $OABC$ in the argand plane, where $O$ is the origin and $A$ be a complex number $z_0$ . Then the equation of the circle that can be inscribed in this square is (Vertices of the square are given in anticlockwise order and $i^2=-1$)

Find the point of intersection of the curves $\arg \left(z-4i\right)=\frac{3\mathrm{\pi }}{4} \& \arg \left(3z+1-3i\right)=\frac{\pi }{4}$ (where, $z$ is a complex number)

For a complex number $z$, If $\left|z\right|\ge 2$, then the minimum value of $\left|z+\frac{1}{2}\right|$

If $\left|z-2+i\right|=\left|z\right|\left|\sin \left(\frac{\pi }{4}-\mathrm{argz}\right)\right|,$ then evaluate the locus of $z$?

In a complex plane the points $A$ and $B$ are at ${\mathrm{z}}_1=5-2\mathrm{i}$ and ${\mathrm{z}}_2=1+\mathrm{i}$. If $\mathrm{P}\left(\mathrm{z}\right)$ moves such that $\left|z-z_1\right|=2\left|z-z_2\right|,$ then the maximum area of the triangle $\mathrm{PAB}$ is _____
 

If the point $2+i$ moves $1$ unit eastwards in the argand plane, then $2$ units northwards and finally from there $2\sqrt{2}$ units in the south - west wards direction. Then, which of the following point represents its new position in the argand plane?