A curve $C$ has parametric equations ${x={{\mathrm{e}}^{t}}+\sin t}$, ${y={{\mathrm{e}}^{t}}-\cos t}

Question

A curve $C$ has parametric equations ${x={{\mathrm{e}}^{t}}+\sin t}$, ${y={{\mathrm{e}}^{t}}-\cos t}$. Describe the shape of $C$ as $t	o \infty $. [2] Find the Cartesian equation of the normal to $C$ at the point $P\left( {{\mathrm{e}}^{	heta }}+\sin 	heta ,\ {{\mathrm{e}}^{	heta }}-\cos 	heta ight)$, where $	heta >0$, giving your answer in the form $y=mx+c$. [3] The normal to $C$ at $P$ meets the $y$-axis at the point $D$, and the curve $C$ meets the positive $x$-axis at the point $E$ that has integral coordinates. Find the coordinates of $D$ and $E$. [3] Find the parametric equations of the mid-point of $DE$ as ${	heta }$ varies. [2]

Accepted Answer

(i) As $t\to\infty$, $C$ tends to infinity in the direction $y=x$, with $|y-x|\le\sqrt2$ (the source's $t\to-\infty$ limit gives $x^2+y^2=1$) (ii) $y=-\frac{e^\theta+\cos\theta}{e^\theta+\sin\theta}x+2e^\theta$ (iii) $D=(0,2e^\theta)$, $E=(1,0)$ (iv) $x=\frac12$, $y=e^\theta$, $\theta>0$

2015 HCI P1 Q6

2015 HCI P1 Q6

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