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 $z=x+iy$ be a complex number $(x, y\in R)$. Let $\mathrm{A}$ and $\mathrm{B}$ be two sets such that $A=\{z:|z|\le 2\}$ and $B=\{z:(z+2y)+\overline{z}\ge 4\}$ the area of region $A\cap B$ is

If $z$ is a complex number, the curves $\left|z\right|=1, |z-2|=1$ and $|z-1|=0$ have a common point at

The locus of a point $z$ represented by the equation $\left|z-1\right|=|z-i|$ on the argand plane is (where, $z\in C, i=\sqrt{-1}$)

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 _____
 

The complex numbers  $\mathrm{z} = \mathrm{x} + \mathrm{iy}$ which satisfy the equation $\left|\frac{\mathrm{z}-5\mathrm{i}}{\mathrm{z}+5\mathrm{i}}\right|=1$, lie on

The locus of the point representing the complex number z for which ${\left|z+3\right|}^2-{\left|z-3\right|}^2=15$ is

The equation of straight line passing through origin and perpendicular to line joining $A\left(z_1\right)$ and $B\left(z_2\right)$ on the complex plane is equal to :

(where $\lambda$ is real parameter.)

The locus of $z$ satisying the inequality $\left|\frac{z+2i}{2z+i}\right|<1$ where $z=x+iy$, is (where $y\mathit{\in }R$ and $\mathit{i}=\sqrt{-1}$)

If $\left|z\right|=3$, then the points representing the complex numbers $-\mathrm{ }1+4z$ lie on a -

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

Let $z, w$ be two complex numbers such that $\left|z\right|=1$ and $\frac{w-1}{w+1}={\left(\frac{z-1}{z+1}\right)}^2$. Then maximum value of $\left|w+1\right|$ is

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

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