A curve $C$ has equation ${{y}^{3}}-2x{{y}^{2}}+3{{x}^{2}}-3=0$ \({{y}^{3}}-2x{{y}^{2}}+3{{x}^{2

Question

A curve $C$ has equation ${{y}^{3}}-2x{{y}^{2}}+3{{x}^{2}}-3=0$ ${{y}^{3}}-2x{{y}^{2}}+3{{x}^{2}}-3=0$[2] Find the equation of the normal to the curve at the point $P\left( 2,3 \right)$.[2] Given that $C$ meets the $y$-axis at the point $R$ and the normal in (ii) meets the $y$-axis at the point $A$, find the area of triangle $APR$ in the form $a-\sqrt[3]{b}$, where $a$ and $b$ are integers to be determined. [3]

Accepted Answer

(i) $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2y^2-6x}{3y^2-4xy}$ (ii) $\displaystyle y=-\frac{1}{2}x+4$ (iii) $\displaystyle 4-\sqrt[3]{3}\text{ units}^2$, where $a=4$ and $b=3$

2021 ASRJC Promo Q4

2021 ASRJC Promo Q4

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