(A) Given that $|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4| = r$ (say). This implies that the points represented by ${z_1}, {z_2}, {z_3}, {z_4}$ are

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(A) Given that $|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4| = r$ (say). This implies that the points represented by ${z_1}, {z_2}, {z_3}, {z_4}$ are at a constant distance $r$ from the point represented by $z$. By definition,a set of points equidistant from a fixed point (the center $z$) lie on a circle with radius $r$. Therefore,the points ${z_1}, {z_2}, {z_3}, {z_4}$ are concyclic.

Class 11 Mathematics 1 Complex Numbers and Quadratic Equations: Complex Numbers Geometry of complex numbers

Easy

If ${z_1}, {z_2}, {z_3}, {z_4}$ are the affixes of four points in the Argand plane and $z$ is the affix of a point such that $|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4|$,then ${z_1}, {z_2}, {z_3}, {z_4}$ are

A
Concyclic
B
Vertices of a parallelogram
C
Vertices of a rhombus
D
In a straight line

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Class 11

Mathematics Questions

Chapter

1 Complex Numbers and Quadratic Equations: Complex Numbers

Subtopic

Geometry of complex numbers

For any complex number $z$, the minimum value of $|z| + |z - 1|$ is

MediumWBJEE 2009

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Let $A = \{ z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1 \}$ and $B = \{ z \in \mathbb{C} : \arg(\frac{z-1}{z+1}) = \frac{2\pi}{3} \}$. Then $A \cap B$ is

DifficultJEE MAIN 2022

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If a complex number $z$ is such that $\frac{z-2i}{z-2}$ is a purely imaginary number and the locus of $z$ is a closed curve,then the area of the region bounded by that closed curve and lying in the first quadrant is

MediumAP EAMCET 2024

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Let $S = \{z : 3 \le |2z - 3(1 + i)| \le 7\}$ be a set of complex numbers. Then $\min_{z \in S} |z + \frac{1}{2}(5 + 3i)|$ is equal to:

DifficultJEE MAIN 2026

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The points $P$ and $Q$ denote the complex numbers $Z_1$ and $Z_2$ in the Argand plane. $O$ is the origin. If $Z_1 \bar{Z}_2 + \bar{Z}_1 Z_2 = 0$ and $\angle POQ = \theta$,then $\sin \theta = $

EasyAP EAMCET 2017

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If ${z_1}, {z_2}, {z_3}, {z_4}$ are the affixes of four points in the Argand plane and $z$ is the affix of a point such that $|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4|$,then ${z_1}, {z_2}, {z_3}, {z_4}$ are

Similar Questions

For any complex number $z$, the minimum value of $|z| + |z - 1|$ is

Let $A = \{ z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1 \}$ and $B = \{ z \in \mathbb{C} : \arg(\frac{z-1}{z+1}) = \frac{2\pi}{3} \}$. Then $A \cap B$ is

If a complex number $z$ is such that $\frac{z-2i}{z-2}$ is a purely imaginary number and the locus of $z$ is a closed curve,then the area of the region bounded by that closed curve and lying in the first quadrant is

Let $S = \{z : 3 \le |2z - 3(1 + i)| \le 7\}$ be a set of complex numbers. Then $\min_{z \in S} |z + \frac{1}{2}(5 + 3i)|$ is equal to:

The points $P$ and $Q$ denote the complex numbers $Z_1$ and $Z_2$ in the Argand plane. $O$ is the origin. If $Z_1 \bar{Z}_2 + \bar{Z}_1 Z_2 = 0$ and $\angle POQ = \theta$,then $\sin \theta = $

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