The equation of a curve $C$ is $2{{x}^{3}}+{{y}^{3}}-3xy=k$, where $k$ is a constant. Fin

Question

The equation of a curve $C$ is $2{{x}^{3}}+{{y}^{3}}-3xy=k$, where $k$ is a constant. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.[2] It is given that $C$ has a tangent which is parallel to the $y$-axis. Show that the $y$-coordinate of the point of contact of the tangent with $C$ must satisfy $a{{y}^{6}}+b{{y}^{3}}-k=0$, where the constants $a$ and $b$ are to be determined.[3] Hence, find the values of $k$ when the line $x=1$ is a tangent to the curve $C$.[3]

Accepted Answer

(i) $\frac{	ext{d}y}{	ext{d}x}=\frac{y-2{{x}^{2}}}{{{y}^{2}}-x}$ (ii) $a=2$, $b=-2$ (iii) $k=4$

2022 JPJC P1 Q7

2022 JPJC P1 Q7

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