Question

Number of asymptotes of the equation $xy(x+y)+a(y^2+xy-x^2)-10a^{2}y=0$ is

• A. $3$
• B. $2$
• C. $0$
• D. None of these

Answer 1

Answer: (A)

$3$

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

Number of asymptotes of above equation is

(1) For asymptotes equate highest degree to zero

i.e. $x^{2}y+x{y}^2=0$

$xy(x+y)=0$

Either $x=0,y=0orx+y=0$

(2) Add constant

$x=c_{1}y=c_{2}x+y=c_3$

Put in above equation

$x=c_1\Rightarrow c_{1}y(c_1+y)+a(y^2+c_{1}y-{c_1}^2)-10a^{2}y=0y=c_2\Rightarrow c_{2}y(c_2+y)+a(y^2+c_{2}y-{c_2}^2)-10a^{2}y=0xy(x+y)+a(y^2+xy-x^2)-10a^{2}y=0x+y=c_3\Rightarrow xy\left(c_3\right)+a[c_3(x-y)+xy]-10a^{2}y=0$

(3) Equate off highest degree of term is equal to zero

From (2)

$c_1+a=0\Rightarrow c_1=-a\Rightarrow x=c_1=-a\Rightarrow x+a=0$

From (3)

$c_2-a=0\Rightarrow c_2=a\Rightarrow y=c_2=a\Rightarrow y-a=0$

From (4)

$c_3+a=0\Rightarrow c_3=-a\Rightarrow x+y=c_3=a\Rightarrow x+y+a=0$

So equation $1$ has $3$ asymptotes.