2026 RI Promo Q5

2026 RI Promo Q5

Junior College 1
8 marks

A curve \(C\) has parametric equations

\[x=\sin t+\sqrt3\cos t,\qquad y=\sqrt3\sin t+\cos t\quad\text{for}\hspace{0.5em}-\pi<t<\pi.\]

  1. Find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) and the exact coordinates of the stationary point(s) of \(C\).[4]
  2. By considering the expressions for \(x^2+y^2\) and \(xy\), find the Cartesian equation of \(C\).[4]

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Answer:(a) \(\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\sqrt3\cos t-\sin t}{\cos t-\sqrt3\sin t}\); stationary points \((\sqrt3,2),(-\sqrt3,-2)\). (b) \(x^2+y^2-\sqrt3xy=1\), excluding \((-\sqrt3,-1)\).

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