Question

The combined equation of the asymptotes of the hyperbola $2x^2 + 5xy + 2y^2 + 4x + 5y = 0$ is -

• A. $2x^2 + 5 xy + 2y^2 + 4x + 5y + 2 = 0$
• B. $2x^2 + 5 xy + 2y^2 + 4x + 5y - 2 = 0$
• C. $2x^2 + 5 xy + 2y^2 = 0$
• D. None of these

Answer 1

Answer: (A)

$2x^2 + 5 xy + 2y^2 + 4x + 5y + 2 = 0$

Answer 2

The correct option is(A): $2x^{2} + 5 xy + 2y^{2} + 4x + 5y + 2 = 0$.

Let the equation of asymptotes be
$2x^{2} + 5xy + 2y^{2} + 4x + 5y + \lambda = 0.......\left(1\right)$
This equation represents a pair of straight lines,
$\therefore\quad abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0$
$\therefore 4\lambda +25-\frac{25}{2} - 8-\lambda\times \frac{25}{4} = 0$
$\Rightarrow -\frac{9\lambda}{4} +\frac{9}{2} = 0 \Rightarrow \lambda = 2$
Putting the value of $\lambda$ in e $\left(1\right)$, we get
$2x^{2} + 5 xy + 2y^{2} + 4x + 5y + 2 = 0$
this is the equation of the asymptotes.