Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then, the area of the rectangle, whose vertices are the roots of the equation $z {¯ z}^{3} + ¯ z z^{3} = 350$ , is

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Solution

The correct option is A 48
$z {¯ z}^{3} + ¯ z z^{3} = 350$
$\Rightarrow {| z |}^{2} ({¯ z}^{2} + z^{2}) = 350$
$\Rightarrow (x^{2} + y^{2}) (x^{2} - y^{2}) = 175$
Since, $x$ and $y$ are integers.
Therefore, $x^{2} + y^{2} = 25$ & $x^{2} - y^{2} = 7$
or, $x^{2} + y^{2} = 35$ & $x^{2} - y^{2} = 5$
$\Rightarrow x = \pm 4$ & $y = \pm 3$ (We do not get integral solutions for $x$ and $y$ for the second case)
Therefore, the vertices of rectangle are $(4, 3), (4, - 3), (- 4, - 3), (- 4, 3)$ .
Thus, Area $= 48$
Hence, option A is correct.

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Let z=x+iy be a complex number where x and y are integers. Then, the area of the rectangle, whose vertices are the roots of the equation z¯z3+¯zz3=350, is

Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then, the area of the rectangle, whose vertices are the roots of the equation $z {¯ z}^{3} + ¯ z z^{3} = 350$ , is