The curve $C$ has equation ${{x}^{2}}+xy+a{{y}^{2}}=36$, where $a$ is a constant such tha

Question

The curve $C$ has equation ${{x}^{2}}+xy+a{{y}^{2}}=36$, where $a$ is a constant such that $a>\frac{1}{4}$. Find the $x$-coordinates of the points on $C$ where the normal is parallel to the $y$-axis, leaving your answers in terms of $a$.[4] For $a=\frac{5}{2}$, the region $R$ is bounded by $C$ and the line $x=-2$ as shown in the diagram. It is also given that all points in the region $R$ are such that $x\le -\frac{y}{2}$. Find the volume formed when $R$ is rotated completely about the $y$-axis, leaving your answers correct to $2$ decimal places.[4]

Accepted Answer

(i) $x=\frac{6}{\sqrt{4a-1}}$ or $-\frac{6}{\sqrt{4a-1}}$ (ii) $x=\frac{-y-\sqrt{144-9{{y}^{2}}}}{2}$; Required volume $=	ext{542}	ext{.87}$ ($2$ d.p.)

2021 NJC P1 Q8

2021 NJC P1 Q8

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