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

Question

If z, barz,-z,- barz forms a rectangle of area 2 √3 square units, then one such z is (A) (1/2)+√3 i (B) (√5+√3 i/4) (C) (3/2)+(√3 i/2) (D) (√3+√11 i/2)

Tardigrade Admin · Accepted Answer

Let z=x+i y Then, vertices of rectangle are (x, y),(x,-y),(-x,-y),(-x, y) Now, area of rectangle =(2 x)(2 y)=4 x y <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/eb1e88c055a538928c595c802e807c3f-.png" /> It is given that, ∴ 4 x y=2 √3 ⇒ 2 x y=√3 x=(1/2), y=√3 ∴ z=(1/2)+√3 i

Q. If $z, \overset{z}{ˉ}, - z, - \overset{z}{ˉ}$ forms a rectangle of area $23$ square units, then one such $z$ is

$\frac{1}{2} + 3 i$

$\frac{5 + 3 i}{4}$

$\frac{3}{2} + \frac{3 i}{2}$

$\frac{3 + 11 i}{2}$

Let $z = x + i y$
Then, vertices of rectangle are $(x, y), (x, - y), (- x, - y), (- x, y)$
Now, area of rectangle $= (2 x) (2 y) = 4 x y$

It is given that,
$∴ 4 x y = 23$
$\Rightarrow 2 x y = 3$
$x = \frac{1}{2}, y = 3$
$∴ z = \frac{1}{2} + 3 i$

Q. If z,zˉ,−z,−zˉ forms a rectangle of area 23​ square units, then one such z is

21​+3​i

45​+3​i​

23​+23​i​

23​+11​i​