If z, z ,-z,- z forms a rectangle of area 2 3 square units, then one such z is

Correct answer: A. (1)/(2)+ 3 i Let a complex number, z=x+i y z = x -i y Then, vertices of rectangle for z, z ,-z,- z are (x, y),(x,-y),(-x,-y),(-x, y) Now, Area of rectangle =(2 x)(2 y)=4 x y It is given that, Area =2 3 =4 x y 2 x y= 3 x=(1)/(2), y= 3 z=(1)/(2)+ 3 i

BITSAT 2024Paper 2

Exam: BITSAT
Year: 2024
Paper: Paper 2
Subject: Mathematics
Chapter: Complex Number

BITSAT 2024 · Paper 2 · Mathematics · Complex Number

If z, z ,-z,- z forms a rectangle of area 2 3 square units, then...

z, \bar{z},-z,-\bar{z}

forms a rectangle of area

2 \sqrt{3}

square units, then one such

z

Options

A $\frac{1}{2}+\sqrt{3 i}$
B $\frac{\sqrt{5}+\sqrt{3 i}}{4}$
C $\frac{3}{2}+\frac{\sqrt{3 i}}{2}$
D $\frac{\sqrt{3}+\sqrt{11 i}}{2}$

Correct answer

A. $\frac{1}{2}+\sqrt{3 i}$

Step-by-step solution

Let a complex number,

z=x+i y\Rightarrow \bar{z}=\bar{x}-i y

Then, vertices of rectangle for

z, \bar{z},-z,-\bar{z}

are

(x, y),(x,-y),(-x,-y),(-x, y)

Now, Area of rectangle

=(2 x)(2 y)=4 x y

It is given that, Area

=2 \sqrt{3}=4 x y \Rightarrow 2 x y=\sqrt{3}\therefore x=\frac{1}{2}, y=\sqrt{3} \quad \therefore z=\frac{1}{2}+\sqrt{3} i

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