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 $z\bar{z}^3+\bar{z}z^3=350$ is
Step-by-Step Solution
Key Concept: $z\bar{z}^3+\bar{z}z^3=2|z|^2\,\text{Re}(z^2)=2(x^2+y^2)(x^2-y^2)$. This reduces to a Diophantine factorisation of 175.
**Step 1: Simplify the equation**
$z\bar{z}^3+\bar{z}z^3=z\bar{z}(\bar{z}^2+z^2)=|z|^2\cdot2\,\text{Re}(z^2)=2(x^2+y^2)(x^2-y^2)=350$, so $(x^2+y^2)(x^2-y^2)=175$, i.e., $x^4-y^4=175$.
**Step 2: Factor 175 to find integer solutions**
$(x^2-y^2)(x^2+y^2)=175$. Trying $x^2-y^2=7,\;x^2+y^2=25$: gives $x^2=16,\;y^2=9$, so $x=\pm4,\;y=\pm3$ ✓. (Other factor pairs yield non-integer solutions.)
**Step 3: Identify the rectangle and compute area**
The four Gaussian integer roots are $\pm4\pm3i$, forming a rectangle of width $2\times4=8$ and height $2\times3=6$. Area $=8\times6=48$.
Correct Answer: 1