A curve $C$ has equation $5{{y}^{2}}-3xy+3{{x}^{2}}-48=0$, with $y\ge 0$. $C$ cuts the $x$

Question

A curve $C$ has equation $5{{y}^{2}}-3xy+3{{x}^{2}}-48=0$, with $y\ge 0$. $C$ cuts the $x$-axis at points $P$ and $R$. The point $Q\left( x,y \right)$ lies on $C$ as shown on the diagram above. Show that $A$, the area of the triangle $PQR$, is given by $A=4y$.[2] Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.[2] Hence, find the exact value of $x$ for which $A$ has a maximum value.[3] [Note: there is no need to show that $A$ is maximum.] If $x$ increases at a constant rate of $5$ units/s, find the rate of change of $A$ when $x=0$.[3]

Accepted Answer

(i) $A=4y$ (ii) $\frac{\mathrm dy}{\mathrm dx}=\frac{3y-6x}{10y-3x}$ (iii) $x=\sqrt{\frac{48}{17}}$ (iv) $6\text{ units}^2\text{/s}$

2018 EJC Promo Q10

2018 EJC Promo Q10

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