The parametric equations of a curve $C$ are \[x=2\theta+\sin 2\theta,\quad y=2\sin\theta\quad\text

Question

The parametric equations of a curve $C$ are $$x=2\theta+\sin 2\theta,\quad y=2\sin\theta\quad\text{for}\hspace{0.5em}0\leq\theta\leq 2\pi.$$ The graph of $C$ meets the $x$-axis at the origin and the point $Q$. The graph of $C$ crosses the $x$-axis at the point $P$. Sketch the graph of $C$ and determine the exact coordinates of $P$ and $Q$.[4] Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{1}{2}\sec\theta$.[4] Hence explain why the graph of $C$ has no stationary points.[1] The tangent to $C$ at the point where $\theta=\dfrac{1}{3}\pi$ cuts the $x$-axis and the $y$-axis at the points $R$ and $S$ respectively. Find the distance $RS$.[5]

Accepted Answer

(a) $P(2\pi,0)$, $Q(4\pi,0)$ (b)(i) Shown (ii) $\sec\theta\neq 0$ for all $\theta$, so $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ is never $0$ (iii) $RS=\sqrt{2}\left(\dfrac{2\pi}{3}-\dfrac{\sqrt{3}}{2}\right)\approx 1.74$

N2026 P1 Q11

N2026 P1 Q11