The parametric equations of the curve $C$ are $x=\tan \theta -1$, $y=2\sec \theta $, where&nbs

Question

The parametric equations of the curve $C$ are $x=\tan \theta -1$, $y=2\sec \theta $, where $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\theta $, leaving your answer in a single trigonometric function.[2] Find the equation of the tangent to $C$ at the point where it cuts the $y$-axis, leaving your answer in exact form.[3] Find the Cartesian equation of $C$.[1] Find the equations of the asymptotes and state the coordinates of any turning point(s).[3] The region bounded by the tangent found in part (b), the curve $C$ and the line $x=-1$ is rotated completely about the $x$-axis. Calculate the volume obtained.[3]

Accepted Answer

(a) $2\sin 	heta $ (b) $y=\sqrt{2}x+2\sqrt{2}$ (c) ${{\left( x+1 ight)}^{2}}+1={{\left( \frac{y}{2} ight)}^{2}}$, $y\ge 2$ (d) Since the centre of the hyperbola is $\left( -1,0 ight)$ and $y\ge 2$, the turning point is $\left( -1,2 ight)$. (e) $2.09$ unit$^{3}$

2024 HCI Promo Q12

2024 HCI Promo Q12

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