Question

Asymptote(s) of the curve $xy(x+y)+a(y^2+xy-x^2)-10a^{2}y=0$ is/are

• A. $x+a=0$
• B. $y-a=0$
• C. $x-a=0$
• D. $y+a=0$

Answer 1

Answer: (A), (B)

The correct options are
A $x+a=0$
B $y-a=0$

$xy(x+y)+a(y^2+xy-x^2)-10a^{2}y=0$

$\Rightarrow (y-a)x^2+(x+a)y^2+ya(x-10a)=0$

Assuming the asymptotes to be $y=mx+c$

Substituting

$\Rightarrow (mx+c-a)x^2+(x+a){(mx+c)}^2+(mx+c)a(x-10a)=0$

For finding the coefficients putting the coefficients equal to zero

$m+m^2=0$

$m=-1,0$

and $(c-a)+2mc+am=0$

For $m=0$

$c=a$

For $m=-1$

$c-a-2c-a=0$

$c=-2a$

For any vertical asymptotes dividing equation by $m^2$

$\Rightarrow \frac{x^2}{m^2}(mx+c-a)+\frac{(x+a)}{m^2}{(mx+c)}^2+\frac{a}{m^2}(mx+c)(x-10a)=0$

$m\to \infty$

$x+a=0$

So asymptotes are

$x+a=0$

$y-a=0$

$y+x+2a=0$