The point $Q\left( x,y \right)$ lies on the curve $4{{x}^{2}}+xy+{{y}^{2}}=36$, where \(y\g

Question

The point $Q\left( x,y \right)$ lies on the curve $4{{x}^{2}}+xy+{{y}^{2}}=36$, where $y\ge 0$, as shown in the diagram above. The curve cuts the $x$-axis at the points $P$ and $R$. Show that $A$, the area of triangle $PQR$, is given by $A=3y.$[3] Find $\frac{\mathrm{d}y}{\mathrm{d}x}$in terms of $x$ and $y$.[2] Hence find the value of $x$ for which $A$ has a stationary value. Using the second derivative test, determine the nature of this stationary value.[6] If $x$ increases at a constant rate of $8$ units/s, find the rate of change of $A$ when $x=0$.[3]

Accepted Answer

Decreasing at a rate of $12 \text{ units}^2\text{/s}$

2013 DHS P1 Q11

2013 DHS P1 Q11

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