Geometry of Complex Numbers

Find the midpoint of the complex numbers $z=2+3i, w=5+7i$.

Find the midpoint of the complex numbers $z=3+9i, w=5+17i$.

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 $i^2=-1$)

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

Cotyledons are also called-

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

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

Consider that $i= \sqrt{-1}$ and $\left|z-2\right|+\left|z-2i\right|=2\sqrt{2}$ . If $\arg \left(z\right)=\frac{\pi }{4},$ then the value of $Re\left(z\right)$ is

The centre of a regular polygon of $n$ sides is located at the point $z=0$  and one of its vertex $z_1$ is known. If $z_2$ be the vertex adjacent to $z_1$ then $z_2$ is equal to

Let $A_0{A}_1{A}_2{A}_3{A}_4{A}_5$ be a regular hexagon inscribed in a circle of unit radius. Then the lengths of the line segment $A_0{A}_1, A_0{A}_2$ and $A_0{A}_4$ is -

The solution of the equation $z^4+4z^{3}i-6z^2-4zi-i=0$ are the vertices of a convex polygon in the complex plane. The area of the polygon is :

The equation of tangent drawn to a circle $\left|z\right|=r$ at the point  $A\left(z_0\right)$ is

A regular hexagon is drawn with two of its vertices forming a shorter diagonal at $\mathrm{z}=-2$ and $\mathrm{z}=1-\mathrm{i}\sqrt{3}$ . The other four vertices are

Equation of tangent drawn to circle $\left|z\right|=r$ at the point $A\left(z_0\right)$ is