<p>If <i>z</i><sub>1</sub> and <i>z</i><sub>2</sub> be the <i>n</i>th root of unity which subtend a right angle at the origin, then <i>n</i> must be of the form</p>
Step-by-Step Solution
Key Concept: The argument of the ratio of two complex numbers determines the angle they subtend at the origin. For a right angle, this argument must equal π/2.
<p><strong>Solution:</strong> The <i>n</i>th roots of unity are given by $e^{2\pi i r/n}$, where $r = 0, 1, 2, \ldots, (n-1)$.</p><p>Let $z_1 = e^{2\pi i r_1/n}$ and $z_2 = e^{2\pi i r_2/n}$, where $0 \leq r_1, r_2 < n$ and $r_1 \neq r_2$.</p><p>Since the line segment joining points with affixes $z_1$ and $z_2$ subtends a right angle at the origin:</p><p>$\arg\left(\frac{z_1}{z_2}\right) = \frac{\pi}{2}$</p><p>$\frac{2\pi r_1}{n} - \frac{2\pi r_2}{n} = \frac{\pi}{2}$</p><p>$\frac{2\pi(r_1 - r_2)}{n} = \frac{\pi}{2}$</p><p>$n = 4(r_1 - r_2)$</p><p>Therefore, $n$ must be of the form $4k$ where $k = r_1 - r_2$.</p><p>∴ Answer is (d).</p>
Correct Answer: d