Let a , b ∈ R and a 2 + b 2 ≠ 0 . Suppose S = z ∈ C : z = 1 a + i b t , t ∈ R , t ≠ 0 , where i = - 1 . If z = x + i y and z ∈ S , then x , y lies on

The y - axis for a = 0 , b ≠ 0

The circle with radius - 1 2 a and centre - 1 2 a , 0 for a < 0 , b ≠ 0

Show that the points in the Argand plane represented by the complex numbers - 2 + 7 i . - 3 2 + 1 2 i , 4 - 3 i , 7 2 1 + i are the vertices of a rhombus.

Given, - 2 + 7 i , - 3 2 + 1 2 i , 4 - 3 i , 7 2 1 + i Let the four points represented by the complex numbers be A - 2 , 7 , B - 3 2 , 1 2 , C 4 , - 3 , D 7 2 , 7 2 The distance between the given points are, A B = - 2 + 3 2 2 + 7 - 1 2 2 = 1 4 + 169 4 = 170 2 units B C = 4 + 3 2 2 + - 3 - 1 2 2 = 121 4 + 49 4 = 170 2 units C D = 4 - 7 2 2 + - 3 - 7 2 2 = 1 4 + 169 4 = 170 2 units A D = - 2 - 7 2 2 + 7 - 7 2 2 = 121 4 + 49 4 = 170 2 units So, all sides are equal, A B = B C = C D = A D . Length of diagonal A C = - 2 - 4 2 + 7 + 3 2 = 36 + 100 = 136 units Length of diagonal B D = - 3 2 - 7 2 2 + 1 2 - 7 2 2 = 100 4 + 36 4 = 136 4 units We know that the diagonals of a rhombus are not equal. ∴ A B C D is a rhombus.

If z is a complex number such that z ≥ 2 , then the minimum value of z + 1 2 :

Is strictly greater than 3 2 but less than 5 2

Show that the points in the Argand diagram represented by the complex numbers 2 + 2 i , - 2 - 2 i , - 2 3 + 2 3 i are the vertices of an equilateral triangle.

Given, 2 + 2 i , - 2 - 2 i , - 2 3 + 2 3 i Let A ( 2 , 2 ) , B ( - 2 , - 2 ) , C ( - 2 3 , 2 3 ) be points represents given complex numbers in the Argand plane. A B = ( 2 + 2 ) 2 + ( 2 + 2 ) 2 ⇒ A B = 4 2 B C = ( - 2 + 2 3 ) 2 + ( - 2 - 2 3 ) 2 ⇒ B C = 4 + 12 - 8 3 + 4 + 12 + 8 3 ⇒ B C = 4 2 A C = ( 2 + 2 3 ) 2 + ( 2 - 2 3 ) 2 ⇒ A C = 4 + 12 + 8 3 + 4 + 12 - 8 3 ⇒ A C = 4 2 ∴ A B = A C = B C So, △ A B C is equilateral.

EASY

Earn 100

Verify whether the points $(2 + 3 i)$ , $(3 - 4 i)$ , $(1 + 2 i)$ and $(3 + 2 i)$ in the complex plane are the vertices of a parallelogram or not

Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

z

is a non-real complex number, then the minimum value of

\frac{I m z^{5}}{{(I m z)}^{5}}

is (Where

I m z =

Imaginary part of

z

)

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Let

a, b \in R

and

a^{2} + b^{2} \neq 0

. Suppose

S = \{z \in C : z = \frac{1}{a + i b t}, t \in R, t \neq 0\},

where

i = \sqrt{- 1} .

z = x + i y

and

z \in S,

then

(x, y)

lies on

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Let

z_{1}

and

z_{2}

be two complex numbers satisfying

| z_{1} | = 9

and

| z_{2} - 3 - 4 i | = 4

. Then the minimum value of

|z_{1} - z_{2}|

is :

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

The equation

I m (\frac{i z - 2}{z - i}) + 1 = 0, z \in C, z \neq i

represents a part of a circle having radius equal to :

EASY

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

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>Geometric Representation of Complex Number

Let

z \in C,

the set of complex numbers. Then the equation,

2 |z + 3 i| - |z - i| = 0

represents:

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Let

z_{1}

and

z_{2}

be any two non-zero complex numbers such that

3 |z_{1}| = 4 |z_{2}| .

z = \frac{3 z_{1}}{2 z_{2}} + \frac{2 z_{2}}{3 z_{1}}

then maximum value of

|z|

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Suppose

z \in C

has argument

θ

such that

0 < θ < \frac{π}{2}

and satisfy the equation

| z - 3 i | = 3

, then what is the value of

\cot θ - \frac{6}{z} ?

EASY

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Let

z_{1}, z_{2}

be the roots of the equation

z^{2} + a z + 12 = 0

and

z_{1}, z_{2}

form an equilateral triangle with origin. Then, the value of

|a|

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Show that the points in the Argand plane represented by the complex numbers $- 2 + 7 i . \frac{- 3}{2} + \frac{1}{2} i, 4 - 3 i, \frac{7}{2} (1 + i)$ are the vertices of a rhombus.

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

z

is a complex number such that

|z| \geq 2,

then the minimum value of

|z + \frac{1}{2}|

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

z_{1}, z_{2}

and

z_{3}

represent the vertices of an equilateral triangle such that

|z_{1}| = |z_{2}| = |z_{3}|

then

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Let

S

be the set of all complex numbers

z

satisfying

|z - 2 + i| \geq \sqrt{5} .

If the complex number

z_{0}

is such that

\frac{1}{|z_{0} - 1|}

is the maximum of the set

\{\frac{1}{|z - 1|} : z \in S\},

then the principal argument of

\frac{4 - z_{0} - \bar{z_{0}}}{z_{0} - \bar{z_{0}} + 2 i}

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Let complex numbers

α

and

\frac{1}{\bar{α}}

lie on circles

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2}

, and

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = 4 r^{2}

, respectively. If

z_{0} = x_{0} + i y_{0}

satisfies the equation

2 {|z_{0}|}^{2} = r^{2} + 2

, then

|α| =

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

The value of

λ

for which the loci

\arg z = \frac{π}{6}

and

| z - 2 \sqrt{3} i | = λ

on the argand plane touch each other is

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Let the point

P

represent

z = x + i y, x, y \in R in

the Argand plane. Let the curves

C_{1}

and

C_{2}

be the loci of

P

satisfying the conditions

(i)

\frac{2 z + i}{z - 2}

is purely imaginary and

(ii)

Arg (\frac{z + i}{z + 1}) = \frac{π}{2}

, respectively. Then the point of intersection of the curves

C_{1}

and

C_{2},

other than the origin, is

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Z

is a complex number such that

|Z| \leq 2

and

- \frac{π}{3} \leq amp Z \leq \frac{π}{3}

. The area of the region formed by locus of

Z

is (in sq. units)

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

Show that the points in the Argand diagram represented by the complex numbers

2 + 2 i, - 2 - 2 i, - 2 \sqrt{3} + 2 \sqrt{3} i

are the vertices of an equilateral triangle.

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

The points in the Argand plane given by

Z_{1} = - 3 + 5 i, Z_{2} = - 1 + 6 i, Z_{3} = - 2 + 8 i, Z_{4} = - 4 + 7 i

form a

HARD

Mathematics>Algebra>Complex Numbers>Geometric Representation of Complex Number

If the four complex numbers

z, \bar{z}, \bar{z} - 2 Re (\bar{z})

and

z - 2 Re (z)

represent the vertices of a square of side

4

units in the Argand plane, then

|z|

is equal to :

Verify whether the points 2+3i,3-4i,1+2i and 3+2i​ in the complex plane are the vertices of a parallelogram or not

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Verify whether the points $(2 + 3 i)$ , $(3 - 4 i)$ , $(1 + 2 i)$ and $(3 + 2 i)$ in the complex plane are the vertices of a parallelogram or not