Complex Numbers
Equations and Roots
Grade 11
Question:
<p>The equation \(z^2 - i|z - 1|^2 = 0\), where \(i = \sqrt{-1}\), has which of the following properties?</p>
<p>(a) no real root</p>
<p>(b) no purely imaginary root</p>
<p>(c) all roots inside \(|z| = 1\)</p>
<p>(d) atleast two roots</p>
Step-by-Step Solution
Key Concept: Separate real and imaginary parts of the complex equation, then analyze the conditions for different forms of $z$.
<p><strong>Method:</strong> Let $z = x + iy$. Then $(x + iy)^2 - i|x + iy - 1|^2 = 0$</p><p>$$x^2 - y^2 + 2ixy - i((x-1)^2 + y^2) = 0$$</p><p><strong>Comparing real and imaginary parts:</strong></p><p>Real part: $x^2 - y^2 = 0$ → $x = \pm y$</p><p>Imaginary part: $2xy - ((x-1)^2 + y^2) = 0$</p><p><strong>Case I:</strong> When $y = x$, we get $2x^2 = (x-1)^2 + x^2$, which simplifies to $0 = -2x + 1$, giving $x = \frac{1}{2}$. This yields $z = \frac{1+i}{2}$ with $|z| = \frac{1}{\sqrt{2}} < 1$.</p><p><strong>Case II:</strong> When $y = -x$, the imaginary equation becomes $-2x^2 = (x-1)^2 + x^2$, which is never satisfied for real $x$.</p><p><strong>Conclusion:</strong> The equation has no real roots (a) ✓, no purely imaginary roots (b) ✓, and all roots lie inside $|z| = 1$ (c) ✓</p>
Correct Answer: a,b,c