Question

Find the asymptotes of the curve x^3+y^3-3axy=0

Answer 1

$here \: is \: your \: answer$

Given
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x³+y³-3axy = 0......................(1)

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✔ The coefficient of the highest power x³ of x is 1 which is a constant .

so, there is no asymptote parallel to x- axis

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✔ The coefficient of the highest power y³ of y is 1, which is a constant.

so, there is no asymptote parallel to y-axis.

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Now ,

Putting x=1 ,y= m in the 3rd, 2nd, 1st degree terms of equation (1)

0³(m)= x³ + y³

0³(m) = 1+m³

O²(m) = -3axy

0²(m) = -3am

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Now ,

For slope O³(m) = 0

1+m³= 0

m= -1

$c = \frac{ - {0}^{2}(m) }{ {0 {}^{1} }^{3}(m) } = \frac{3am}{3 {m}^{2} } \\ \\ c = \frac{a}{m} = \frac{a}{ - 1} = - a$
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where m= -1 , c= -a

so, y= mx+c

y= -x-a = -(x+a) is the asymptote of the given curve .
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$hope \: this \: helps \: you$