    Question

# If $z=x+iy$ is a complex number where $x$ and $y$ are integers. Then, the area of the rectangle whose vertices are the roots of the equation $z{\overline{z}}^{3}+\overline{z}{z}^{3}=350$ is

A

$48$

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B

$32$

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C

$40$

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D

$80$

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Solution

## The correct option is A $48$Explanation for the correct option.Step 1. Form the equations.For a complex number $z=x+iy$, the conjugate is given as: $\overline{z}=x-iy$.Now simplify the equation $z{\overline{z}}^{3}+\overline{z}{z}^{3}=350$.$z{\overline{z}}^{3}+\overline{z}{z}^{3}=350\phantom{\rule{0ex}{0ex}}⇒z\overline{z}\left({\overline{z}}^{2}+{z}^{2}\right)=350\phantom{\rule{0ex}{0ex}}⇒\left(x+iy\right)\left(x-iy\right)\left[{\left(x-iy\right)}^{2}+{\left(x+iy\right)}^{2}\right]=350\phantom{\rule{0ex}{0ex}}⇒\left({x}^{2}-{i}^{2}{y}^{2}\right)\left[{x}^{2}+{i}^{2}{y}^{2}-2xyi+{x}^{2}+{i}^{2}{y}^{2}+2xyi\right]=350\phantom{\rule{0ex}{0ex}}⇒\left({x}^{2}+{y}^{2}\right)2\left({x}^{2}-{y}^{2}\right)=350\phantom{\rule{0ex}{0ex}}⇒\left({x}^{2}+{y}^{2}\right)\left({x}^{2}-{y}^{2}\right)=175\phantom{\rule{0ex}{0ex}}⇒\left({x}^{2}+{y}^{2}\right)\left({x}^{2}-{y}^{2}\right)=25×7$Now, as $x,y\in R$, so sum of squares would be greater than difference of squares. So,$\left\{\begin{array}{l}{x}^{2}+{y}^{2}=25...\left(1\right)\\ {x}^{2}-{y}^{2}=7...\left(2\right)\end{array}\right\$Step 2. Solve the equations.Add the equations $1$ and $2$.${x}^{2}+{y}^{2}+{x}^{2}-{y}^{2}=25+7\phantom{\rule{0ex}{0ex}}⇒2{x}^{2}=32\phantom{\rule{0ex}{0ex}}⇒{x}^{2}=16\phantom{\rule{0ex}{0ex}}⇒x=±4$Substitute ${x}^{2}=16$ in equation $1$.$16+{y}^{2}=25\phantom{\rule{0ex}{0ex}}⇒{y}^{2}=9\phantom{\rule{0ex}{0ex}}⇒y=±3$So, the coordinates of the vertices of the rectangle are $\left(4,3\right),\left(4,-3\right),\left(-4,3\right),\left(-4,-3\right)$.Step 3. Find the area of the rectangle.The length of the rectangle whose vertices are $\left(4,3\right),\left(4,-3\right),\left(-4,3\right),\left(-4,-3\right)$ is $4-\left(-4\right)=8$ and its breadth is $3-\left(-3\right)=6$.So, the area of the rectangle is $\begin{array}{rcl}\mathrm{Area}& =& \mathrm{Length}×\mathrm{Breadth}\\ & =& 8×6\\ & =& 48\end{array}$Hence, the correct option is A.  Suggest Corrections  0      Similar questions
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