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

Let $z_1, z_2, z_3$ be three complex numbers such that $arg\left(\frac{z_3-z_2}{z_1-z_2}\right)=\frac{\mathrm{\pi }}{6}$ and $\left|z_1-4\right|=\left|z_2-4\right|=\left|z_3-4\right|$, then the value of ${z_1}^2+{z_3}^2-4z_1-4z_3-z_1{z}_3+18$ is -

If ${\log }_{\frac{1}{2}}\left(\frac{\left|z-1\right|+4}{3\left|z-1\right|-2}\right)>1$, then the locus of $z$ is exterior to circle whose

For a complex number $z$, if $\lambda$ and $\mu$ are the greatest and least distance between the curves $\left|z\right|=2$ and $\left|z-5-12i\right|=2$ respectively, then the value of ${\lambda }^2+{\mu }^2$ is

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

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 $i^2=-1$)

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

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 $\left|z-1\right|=1$ and $\omega =Re\left(z+2\right),$ then $\omega$ lies on

If $z_1$ and $z_2$ are two complex numbers satisfying the equation $\left|z-1\right|+\left|z+5\right|=6$ and $\left|z+2\right|=2$ then

Each of the circles $\left|z-1-i\right|=1$ and $\left|z-1+i\right|=1$ touches internally a circle of radius 2. The equation of the circle touching all the three circles can be

Let $A=\left\{z\in C: \left|z\right|=25\right\}$ and $B=\left\{z\in C: \left|z+5+12i\right|=4\right\}.$ Then the minimum value of $\left|z-\omega \right|,$ for $z\in A$ and $\omega \in B,$ is:

The perimeter of the locus represented by $\mathrm{a}\mathrm{r}\mathrm{g}\left(\frac{z+i}{z-i}\right)=\frac{\pi }{4}$ is equal to

If the imaginary part of the expression $\frac{z-1}{e^{\theta i}}+\frac{e^{\theta i}}{z-1}$ be zero, then locus of $z$ is $\left(\mathrm{where} i=\sqrt{-1}\right)$

If the complex number $z$ lies on a circle with centre at the origin and radius $\frac{1}{4},$ then the complex number $-1+8z$ lies on a circle with radius