Question

Find all the asymptotes of the curve,

x³+y³ = 3axy, where a>0.

please solve it very urgent

Answer 1

Let's find a slant asymptote of this curve

Let's rewrite the equation

$x^3+y^3-3axy=0$

The slant asymptote is $y=mx+c$

Substuting this to the equation will give us

$x^3+(mx+c)^3-3ax(mx+c)=(1+m^2)x^3+(3m^2c+3am)x^2+(3mc^2-3ac)x+c^3=0$

Therefore

$1+m^3=0;\qquad m=-1$

$3m^2c-3am=0;\qquad mc-a=0;\qquad c=-a$

Hence $y=-x-a$ is a slant asymptote of this curve