Rectangle $OPQR$ has sides parallel to the coordinate axes, with $O$ at the origin, $P$ on the

Question

Rectangle $OPQR$ has sides parallel to the coordinate axes, with $O$ at the origin, $P$ on the positive $x$-axis and $R$ on the positive $y$-axis. The vertex $Q(x,y)$ lies in the first quadrant on the curve $C$, whose equation is $x^2+2xy+4y^2=52$. The variables $x$ and $y$ are measured in units. Let $A$ be the area of the rectangle. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.[2] Hence, show that $A$ has a stationary value when $x=2y$, and find the exact maximum value of $A$.[4] At an instant, $Q=(6,1)$ and $x$ is increasing at a constant rate of $\frac{5}{2}$ units/s. Find the rate of change of $A$ at this instant.[3]

Accepted Answer

(i) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{x+y}{x+4y}$ (ii) $A_{\max}=\frac{26}{3}\text{ units}^2$ (iii) $\frac{\mathrm{d}A}{\mathrm{d}t}=-8\text{ units}^2\text{/s}$ (decreasing)

Rectangle under an implicit curve

Rectangle under an implicit curve

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