Question

Let \( z = x + iy \) be a complex number with \( x, y \in \mathbb{Z} \). Then the area (in square units) of the rectangle whose vertices are the roots of the equation \( \bar{z} \cdot z^3 + z \cdot \bar{z}^3 = 350 \) is:

• A. \( 48 \)
• B. \( 32 \)
• C. \( 40 \)
• D. \( 44 \)

Hint:

Symmetric Complex Expressions}
Use \( z = x + iy \), \( \bar{z} = x - iy \)
Use symmetry and power expansions like \( z^n \), \( \bar{z}^n \)
Such equations often relate to geometrical structures

Answer 1

The Correct Option is A

Let \( z = x + iy \), then \( \bar{z} = x - iy \). We compute: \[ \bar{z} \cdot z^3 + z \cdot \bar{z}^3 = 350 \] Using identities and expressing real and imaginary parts, this simplifies to an expression in \( x \) and \( y \). After solving, we find that the expression corresponds to a symmetric quartic that simplifies to an equation of a rectangle's diagonal structure. Solving leads to area = \( 48 \) square units.