A curve $D$ has equation ${{x}^{2}}+2xy+3{{y}^{2}}=16$, where $y0$. Show that $\frac{\mathrm{d}y}{\

Question

A curve $D$ has equation ${{x}^{2}}+2xy+3{{y}^{2}}=16$, where $y>0$. Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{-x-y}{x+3y}$. [2] Find the equation of the tangent to $D$ at the point where $x=-4$. [3] The diagram above shows a right-angled triangle $ABC$. The lengths $BC$, $AC$ and $AB$ are denoted by $a$cm, $b$cm, and $c$cm respectively and the triangle $ABC$ has a fixed perimeter of $30$cm. Show that $450-30a-30b+ab=0$. [2] Given that $AC$ is increasing at a constant rate of $6$cm per second, find the rate of change of $BC$ when $b=12$. [3]

Accepted Answer

$\frac{dy}{dx} = \frac{-x-y}{x+3y}$; $y = \frac{1}{3}x + 4$; BC decreases at $\frac{25}{3}$ cm/s

2022 EJC J2 MYE P1 Q8

2022 EJC J2 MYE P1 Q8

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