If $w=\frac{z}{z-\frac{1}{3}i}$ and $\left|w\right|=1$, then $z$ lies on

Important Questions on Complex Numbers and Quadratic Equations

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

If $z_1,z_2$ and $z_3$ represent the vertices of an equilateral triangle such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|$ then

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

The equation $|z-i|=|z-1|,i=\sqrt{-1},$ represents:

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

If the four complex numbers $z,\overline{z},\overline{z}-2Re\left(\overline{z}\right)$ and $z-2Re\left(z\right)$ represent the vertices of a square of side $4$ units in the Argand plane, then $\left|z\right|$ is equal to :

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

$Z$ is a complex number such that $\left|Z\right|\le 2$ and $-\frac{\pi }{3}\le ampZ\le \frac{\pi }{3}$. The area of the region formed by locus of $Z$ is (in sq. units)

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

The equation $Im\left(\frac{iz-2}{z-i}\right)+1=0, z\in C, z\ne i$ represents a part of a circle having radius equal to :

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

Suppose $z\in C$ has argument $\theta$ such that $0<\theta <\frac{\pi }{2}$ and satisfy the equation $|\mathrm{z}-3\mathrm{i}|=3$,  then what is the value of $\cot \theta -\frac{6}{z}?$

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z-2+i\right|\ge \sqrt{5}.$ If the complex number $z_0$ is such that $\frac{1}{\left|z_0-1\right|}$ is the maximum of the set $\left\{\frac{1}{\left|z-1\right|}:z\in S\right\},$ then the principal argument of $\frac{4-z_0-\overline{z_0}}{z_0-\overline{z_0}+2i}$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

If $z_1$ and $z_2$ be two non-zero complex numbers such that $\frac{z_1}{z_2}+\frac{z_2}{z_1}=1,$ then the origin and the points represented by $z_1$ and $z_2$

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

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

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

The point represented by $2+i$ in the Argand plane moves $1$ unit eastwards, then $2$ units northwards and finally from there $2\sqrt{2}$ units in the south-west wards direction. Then its new position in the Argand plane is at the point represented by : 

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

The points in the Argand plane given by $Z_1=-3+5i, Z_2=-1+6i, Z_3=-2+8i, Z_4=-4+7i$ form a

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

The largest value of $r$, for which the region represented by the set $\left\{\omega \in C|\left|\omega -4-i\right|\le r\right\}$ is contained in the region represented by the set $\left\{z\in C|\left|z-1\right|\le \left|z+i\right|\right\}$, is equal to :

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

Let the point $P$ represent $z=x+iy, x, y\in R \mathrm{in}$ the Argand plane. Let the curves ${\mathrm{C}}_1$ and $C_2$ be the loci of $P$ satisfying the conditions $\left(\mathrm{i}\right)$ $\frac{2z+i}{z-2}$ is purely imaginary and $\left(\mathrm{ii}\right)$ $Arg\left(\frac{z+i}{z+1}\right)=\frac{\pi }{2}$, respectively. Then the point of intersection of the curves $C_1$ and $C_2,$ other than the origin, is

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

The value of $\lambda$ for which the loci $\arg z=\frac{\pi }{6}$ and $|z-2\sqrt{3}i|=\lambda$ on the argand plane touch each other is 

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

Let $A\left(3-i\right),B\left(2+i\right)$ be two points in the Argand plane. If the point $P$ represents the complex number $z=x+iy,$ which satisfies $\left|z-3+i\right|=\left|z-2-i\right|,$ then the locus of the point $P$ is

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

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

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

The equation $z\overline{z}+(2-3i)z+(2+3i)\overline{z}+4=0$ represents a circle of radius

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

Let $z\in C$ be such that $\left|z\right|<1.$ If $\omega =\frac{5+3z}{5\left(1-z\right)},$ then:

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

Let $z\in C,$ the set of complex numbers. Then the equation, $2\left|z+3i\right|-\left|z-i\right|=0$ represents:

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Geometry of Complex Numbers

Let $z_1$ and $z_2$ be two complex numbers satisfying $\left|z_1\right|=9$ and $|z_2-3-4i|=4$. Then the minimum value of $\left|z_1-z_2\right|$ is :