A curve $C$ has parametric equations $x=1+{{\mathrm{e}}^{t}}$, $y=t-{{\mathrm{e}}^{t}}$ for $t0$. S

Question

A curve $C$ has parametric equations $x=1+{{\mathrm{e}}^{t}}$, $y=t-{{\mathrm{e}}^{t}}$ for $t>0$. Sketch the graph of $C$. [2] Find the equation of the tangent to $C$ at the point $P\left( 1+{{\mathrm{e}}^{p}},p-{{\mathrm{e}}^{p}} \right)$ in the form $y=ax+b$, where $a$ and $b$ are in terms of $p$. [3] The tangent to $C$ at $P$ meets the $y$-axis at the point $Q$. Point $R$ lies on another curve such that $P$ is the midpoint of $QR$. Find parametric equations of the curve traced by $R$ as $p$ varies. [4]

Accepted Answer

(ii) $y=(e^{-p}-1)x-e^{-p}+p$; (iii) $x=2+2e^p$, $y=p-2e^p+e^{-p}$, $p>0$

2020 TMJC P2 Q2

2020 TMJC P2 Q2

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