Question

find the asymptote for the equation
${x}^{2} y - y - x = 0$
​

Answer 1

xy(x+y)+a(y

2

+xy−x

2

)−10a

2

y=0

⇒(y−a)x

2

+(x+a)y

2

+ya(x−10a)=0

Assuming the asymptotes to be y=mx+c

Substituting

⇒(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

⇒

m

2

x

2

(mx+c−a)+

m

2

(x+a)

(mx+c)

2

+

m

2

a

(mx+c)(x−10a)=0

m→∞

x+a=0

So asymptotes are

x+a=0

y−a=0

y+x+2a=0