A curve $C$ has parametric equations $x={{\mathrm{e}}^{u}}$, $y={{u}^{2}}$, for \(u\in \m

Question

A curve $C$ has parametric equations $x={{\mathrm{e}}^{u}}$, $y={{u}^{2}}$, for $u\in \mathbb{R}$. Sketch $C$, stating the equation of the asymptote and the coordinates of any points where $C$ cuts the axes.[3] Find the equation of the tangent to $C$ at the point where $u=-1$. Leave your answer in the form of $y=mx+c$ where $m$ and $c$ are exact values to be determined.[3] For the rest of the question, all lengths are measured in kilometres (km). The region bounded by $C$ and the line $y=5$ represents a man-made lake. A circle with radius $0.5$ and centre $P\left( 1,2 \right)$ represents a man-made island on the lake. A bridge is built from a point $Q$ on $C$, where $u\le 0$, to a point $R$ on the edge of the circular island such that $QR$ produced passes through point $P$. It is given that the distance between $Q$ and $P$ is denoted by $l$. Show that ${{l}^{2}}={{\mathrm{e}}^{2u}}-2{{\mathrm{e}}^{u}}+{{u}^{4}}-4{{u}^{2}}+5$.[2] Use differentiation to find the values of $u$ which give stationary values of ${{l}^{2}}$.[2] Show that only one of the values of $u$ found in part (iv) gives a minimum value of ${{l}^{2}}$. Hence find the shortest length of the bridge $QR$.[4]

Accepted Answer

(i) Asymptote: $x=0$; Intercept(s): $\left( 1,0 ight)$(ii) $m=-2	ext{e}$, $c=3$ (iii) ${{	ext{e}}^{2u}}-2{{	ext{e}}^{u}}+{{u}^{4}}-4{{u}^{2}}+5$ (iv) $u=-1.39$ or $0$. (v) $QR=0.254$ km

2022 VJC Promo Q12

2022 VJC Promo Q12

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