Complex Numbers
Geometric Properties
Grade None
Question:
<p>If \(A(z_1), B(z_2), C(z_3), D(z_4)\) lies on \(|z| = 4\) (taken in order), where \(z_1 + z_2 + z_3 + z_4 = 0\), then:</p>
<p>(a) Max. area of quadrilateral ABCD = 32</p>
<p>(b) Max. area of quadrilateral ABCD = 16</p>
<p>(c) The triangle DABC is right angled</p>
<p>(d) The quadrilateral ABCD is rectangle</p>
Step-by-Step Solution
Key Concept: The condition $z_1 + z_2 + z_3 + z_4 = 0$ means diagonals bisect at origin; combined with equal moduli, the quadrilateral must be a rectangle.
<p><strong>Step 1:</strong> From $z_1 + z_2 + z_3 + z_4 = 0$, we have $z_1 + z_3 = -(z_2 + z_4)$.</p><p>This means the diagonals AC and BD bisect each other at the origin.</p><p><strong>Step 2:</strong> Since all points lie on circle of radius 4 and diagonals bisect each other at center, ABCD is a cyclic quadrilateral inscribed in a circle with diagonals intersecting at center.</p><p>Such a quadrilateral is a rectangle (since diagonals are equal and bisect each other).</p><p><strong>Step 3:</strong> For a rectangle inscribed in circle of radius 4, with sides $a$ and $b$:</p><p>Diagonal = $2 \times 4 = 8$, so $a^2 + b^2 = 64$.</p><p>Area = $ab$. By AM-GM: $ab \leq \frac{a^2 + b^2}{2} = 32$</p><p>Maximum area = 32 when $a = b = 4\sqrt{2}$ (square).</p>
Correct Answer: a,d