find-the-asymptote-of-folium-of-Descartes-x-3-y-3-3axy-and-a-is-a-constant-gt-0-

Question Number 71739 by Tony Lin last updated on 19/Oct/19

f i n d t h e a s y m p t o t e o f f o l i u m o f

D e s c a r t e s x^{3} + y^{3} = 3 a x y, a n d a i s a

c o n s t a n t > 0

Answered by Tony Lin last updated on 19/Oct/19

l e t y = t x

\Rightarrow x^{3} + {(t x)}^{3} = 3 {a t x}^{2}

{\begin{cases} x = \frac{3 a t}{1 + t^{3}} \\ y = \frac{3 {a t}^{2}}{1 + t^{3}} \end{cases}

x \to \pm \infty \Rightarrow t \to - 1

l e t t h e a s y m p t o t e L : m x + b

\lim_{t \to - 1} \frac{y}{x} = - 1 = m

\lim_{t \to - 1} [y - (- x)]

= \lim_{x \to - 1} \frac{3 a t (t + 1)}{(t + 1) (t^{2} - t + 1)}

= - a

= b

t h e r e f o r e

a s y m p t o t e L : - x - a

Answered by MJS last updated on 19/Oct/19

x^{3} + y^{3} - 3 a x y = 0

turning

x = \frac{\sqrt{2}}{2} (x^{*} + y^{*}) \land y = \frac{\sqrt{2}}{2} (x^{*} - y^{*})

\frac{3 (a + \sqrt{2} x^{*})}{2} {(y^{*})}^{2} - \frac{(3 a - \sqrt{2} x^{*}) {(x^{*})}^{2}}{2} = 0

\Rightarrow y^{*} = \pm \frac{x^{*} \sqrt{9 a - 3 \sqrt{2} x^{*}}}{3 \sqrt{a + \sqrt{2} x^{*}}}

denominator not defined for x^{*} = - \frac{\sqrt{2} a}{2} + 0 y^{*}

turning back

p = \frac{\sqrt{2}}{2} (x + y) \land q = \frac{\sqrt{2}}{2} (x - y)

\frac{\sqrt{2}}{2} (x + y) = - \frac{\sqrt{2} a}{2}

y = - x - a is the asymptote

Commented by MJS last updated on 19/Oct/19

there might be an easier path \dots

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