A curve $C$ has equation ${{x}^{2}}-3{{y}^{2}}=3$. Sketch $C$, giving the equations of asympto

Question

A curve $C$ has equation ${{x}^{2}}-3{{y}^{2}}=3$. Sketch $C$, giving the equations of asymptote(s) and axial intercept(s) in exact form.[3] A complex number is given by $z=x+\mathbf{i}y$, where $x$ and $y$ are real. Given that $\operatorname{Re}\left( z \right)$ is positive, and that $x$ and $y$ satisfy the equation ${{x}^{2}}-3{{y}^{2}}=3$, state the behaviour of the argument of $z$ when $\operatorname{Re}\left( z \right)\to \infty $.[3]

Accepted Answer

(i) Asymptotes $y=\pm\dfrac{x}{\sqrt3}$, axial intercepts $(\pm\sqrt3,0)$ (ii) If $y>0$, $\arg z$ increases from $0$ towards $\dfrac\pi6$, while if $y<0$, it decreases from $0$ towards $-\dfrac\pi6$

2022 HCI P1 Q4

2022 HCI P1 Q4

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