Question

Let $z = x + iy$ be a complex number where $x, y$ are integers then the area of the rectangle whose vertices are roots of the equation $z\bar{z}^3 + \bar{z}z^3 =1088$, is

• A. $15$
• B. $30$
• C. $60$
• D. None of these

Answer 1

Answer: (C)

Given, $z \bar{z}^{3} + \bar{z} z^3 = 1088$
$\Rightarrow z\bar{z}^2 (z^2 + \bar{z}^2) = 1088$
$\Rightarrow z\bar{z}\{(z + \bar{z})^2 -2z \bar{z}\} = 1088$
$\Rightarrow (x^2 + y^2)[(2Re(z))^2 - 2(x^2 + y^2)] = 1088$
$\Rightarrow (x^2 + y^2)(4x^2 - 2x^2 - 2y^2) = 1088$
$( \because z + \bar{z} =2Rez = 2x)$
$\Rightarrow (x^2 + y^2)( x^2 - y^2) = 544 = 34 \times 16$
or $(32 \times 17$ which is impossible)
$\Rightarrow x^2 + y^2 = 34$ and $x^2 - y^2 = 16$
$\Rightarrow (x, y) = (\pm 5, \pm 3) = (5, - 3), (-5, 3), (-5, -3)$ & $(5, -3)$

$\therefore $ Required area (area of rectangle)
$= 4(x \times y) = 4 \times 5 \times 3 = 60$ units