<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1759231539251-cxxbmpx5pke.png" width="400" class="aligncenter" /></figure>
<p>The diagram shows the graph of Folium of Descartes with cartesian equation</p>
<p style="text-align: center">${{x}^{3}}+{{y}^{3}}=3axy$,</p>
<p>where a is a positive constant. The curve passes through the origin, and has an oblique asymptote with equation $y=-x-a$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that $\left( 0,0 \right)$ is a stationary point on the curve, find, in terms of a, the coordinates of the other stationary point.</span></div><p style="text-align: right;">[5]</p></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of</span></div><div class="list-item-paragraph" style="text-align: center">${{\left| x \right|}^{3}}+{{y}^{3}}=3a\left| x \right|y$,</div><div class="list-item-paragraph">including the equations of any asymptotes, coordinates of the stationary points and the point where the graph crosses the $x$-axis.</div><p style="text-align: right;">[3]</p></li></ol>

The diagram shows the graph of Folium of Descartes with cartesian equation

where a is a positive constant. The curve passes through the origin, and has an oblique asymptote with equation $y=-x-a$.

Given that $\left( 0,0 \right)$ is a stationary point on the curve, find, in terms of a, the coordinates of the other stationary point.

${{\left| x \right|}^{3}}+{{y}^{3}}=3a\left| x \right|y$,

including the equations of any asymptotes, coordinates of the stationary points and the point where the graph crosses the $x$-axis.

Concavity of Graphs

Applications of Differentiation

International Baccalaureate Higher Level Mathematics

IB HL Math

2019 RI P1 Q5

$x^{3}+y^{3} = 3axy$. Differentiating: $3x^{2}+3y^{2}\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3a\left[x\dfrac{\mathrm{d}y}{\mathrm{d}x}+y\right]$

$\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{ay - x^{2}}{y^{2} - ax}$

$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0$: $y = \dfrac{x^{2}}{a}$. Sub into curve: $x^{3} + \dfrac{x^{6}}{a^{3}} = 3x^{3}$

$\dfrac{x^{6}}{a^{3}} = 2x^{3} \Rightarrow x^{3} = 2a^{3} \Rightarrow x = 2^{1/3}a$

$y = \dfrac{(2^{1/3}a)^{2}}{a} = 2^{2/3}a$. Other stationary point: $(2^{1/3}a,\; 2^{2/3}a)$.

$|x|^{3}+y^{3} = 3a|x|y$: symmetric about $y$-axis (replace $x$ with $|x|$).

Stationary points: $(\pm 2^{1/3}a,\; 2^{2/3}a)$. Asymptotes: $y = x - a$ and $y = -x - a$.

Crosses $x$-axis at $(0, 0)$. Crosses $y$-axis at $(0, -a)$ (on asymptote branch).

2019 RI P1 Q5

2019 RI P1 Q5

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