Complex Numbers
Modulus and Argument
Grade 11
Question:
<p>Let <span>\(z, w\)</span> be complex numbers such that <span>\(\bar{z}+i\bar{w}=0\)</span> and <span>\(\arg\, zw = \pi\)</span>. Then <span>\(\arg\, z\)</span> equals</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(\dfrac{5\pi}{4}\)</p>
<p>\(\dfrac{3\pi}{4}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
Step-by-Step Solution
Key Concept: From the constraint $\bar{z} + i\bar{w} = 0$, deduce the relationship between $z$ and $w$ by taking conjugates. Then use $\arg(zw) = \pi$ to find the individual argument of $z$.
<p><strong>Step 1:</strong> From $\bar{z} + i\bar{w} = 0$, we get $\bar{z} = -i\bar{w}$.</p><p><strong>Step 2:</strong> Taking conjugate of both sides: $z = -iw$, which means $z = e^{-i\pi/2} \cdot w$.</p><p><strong>Step 3:</strong> Therefore $\arg(z) = \arg(w) - \frac{\pi}{2}$ (modulo $2\pi$).</p><p><strong>Step 4:</strong> Since $\arg(zw) = \pi$, we have $\arg(z) + \arg(w) = \pi$ (choosing the principal branch).</p><p><strong>Step 5:</strong> Substituting $\arg(z) = \arg(w) - \frac{\pi}{2}$: $(\arg(w) - \frac{\pi}{2}) + \arg(w) = \pi$</p><p><strong>Step 6:</strong> $2\arg(w) = \frac{3\pi}{2}$, so $\arg(w) = \frac{3\pi}{4}$.</p><p><strong>Step 7:</strong> Therefore $\arg(z) = \frac{3\pi}{4} - \frac{\pi}{2} = \frac{\pi}{4}$.</p><p>∴ Answer: C</p>
Correct Answer: C