Question

The equation of one of the asymptotes of the hyperbola $x^2+3xy+2y^2+2x+3y=0$ is :

• A. $x+2y-1=0$
• B. $x+2y+1=0$
• C. $x+y-1=0$
• D. $x+y+1=0$

Answer 1

Answer: (B), (D)

The correct options are
B $x+2y+1=0$
D $x+y+1=0$

The combined equation of the asymptoes differ from of the hyperbola only in the constant term.
$\therefore$ the equation of the asympotoes is
$x^2+3xy+2y^2+2x+3y+k=0$ ... $\left(1\right)$
$x^2+3xy+2y^2=(x+2y)(x+y)$
$\therefore$ equation $\left(1\right)$ can be written as
$(x+2y+{\lambda }_1)(x+y+{\lambda }_2)=0$ ... $\left(2\right)$
Equating the coefficient of $x$ and $y$ in $\left(1\right)$ and $\left(2\right)$, we get
${\lambda }_1+{\lambda }_2=2$ and ${\lambda }_1+2{\lambda }_2=3\Rightarrow {\lambda }_1={\lambda }_2=1$
Therefore, the equation are $x+2y+1=0$ and $x+y+1=0$.
Hence, options 'B' and 'D' are correct