A complex number $z$ varies with $t$ such that \(z = 2 \cos t + \mathrm{i}(3 \sin t), \quad \tex

Question

A complex number $z$ varies with $t$ such that $z = 2 \cos t + \mathrm{i}(3 \sin t), \quad \text{where}\hspace{0.5em} 0 \leq t < 2\pi$. By taking $x = \mathrm{Re}(z)$ and $y = \mathrm{Im}(z)$, sketch on an Argand diagram the curve that shows the positions of the points representing the complex number $z$. Find the cartesian equation of this curve.[3] Two complex numbers $z_1$ and $z_2$ for two distinct values of $t$ are such that $|z_1| = |z_2|$ and $0 < \arg(z_1) < \frac{\pi}{2}$. By referring to the Argand diagram in part (a), find the possible values of $\arg(z_1 + z_2)$.[2] It is given further that $z_1$ and $z_2$ are roots to the quadratic equation $z^2 + \alpha z + \beta = 0$. Explain whether it is necessary for $\alpha$ to be real.[2] Given that $\alpha$ is not real and $|z_1| = |z_2| = \frac{\sqrt{26}}{2}$, find the values for $\alpha$ and $\beta$.[4]

Accepted Answer

(a) $\frac{x^2}{4}+\frac{y^2}{9}=1$; ellipse shown. (b) $0$ or $\frac{\pi}{2}$; undefined when $z_2=-z_1$. (c) No. (d) $\alpha=-3\sqrt2\mathrm i,\ \beta=-\frac{13}{2}$.

2025 DHS P1 Q8

2025 DHS P1 Q8

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