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. $48$
• B. $32$
• C. $40$
• D. $80$

Answer 1

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 $z{\overline{z}}^3+\overline{z}{z}^3=350$.

$$\begin{gathered} z{\overline{z}}^3+\overline{z}{z}^3=350 \\ \Rightarrow z\overline{z}\left({\overline{z}}^2+z^2\right)=350 \\ \Rightarrow \left(x+iy\right)\left(x-iy\right)\left[{\left(x-iy\right)}^2+{\left(x+iy\right)}^2\right]=350 \\ \Rightarrow \left(x^2-i^2{y}^2\right)\left[x^2+i^2{y}^2-2xyi+x^2+i^2{y}^2+2xyi\right]=350 \\ \Rightarrow \left(x^2+y^2\right)2\left(x^2-y^2\right)=350 \\ \Rightarrow \left(x^2+y^2\right)\left(x^2-y^2\right)=175 \\ \Rightarrow \left(x^2+y^2\right)\left(x^2-y^2\right)=25\times 7 \end{gathered}$$

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 $2$.

$$\begin{gathered} x^2+y^2+x^2-y^2=25+7 \\ \Rightarrow 2x^2=32 \\ \Rightarrow x^2=16 \\ \Rightarrow x=\pm 4 \end{gathered}$$

Substitute $x^2=16$ in equation $1$.

$$\begin{gathered} 16+y^2=25 \\ \Rightarrow y^2=9 \\ \Rightarrow y=\pm 3 \end{gathered}$$

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

$\begin{array}{rcl}\mathrm{Area}& =& \mathrm{Length}\times \mathrm{Breadth}\\ & =& 8\times 6\\ & =& 48\end{array}$

Hence, the correct option is A.