Question

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

• A. $2x^{2} + 5xy + 2y^{2} = 0$
• B. $2x^{2} + 5xy + 2y^{2} - 4x + 5y + 2 = 0 = 0$
• C. $2x^{2} + 5xy + 2y^{2} + 4x + 5y - 2 = 0$
• D. $2x^{2} + 5xy + 2y^{2} + 4x + 5y + 2 = 0$

Answer 1

Answer: (D)

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

Answer 2

Given, equation of hyperbola

$2x^{2} + 5xy + 2y^{2} + 4x + 5y = 0$ and equation of asymptotes

$2x^{2} + 5xy + 2y^{2} + 4x + 5y + \lambda = 0$ ......(i) which is the equation of a pair of straight lines. We know that the standard equation of a pair of straight lines is

$ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0$Comparing equation (i) with standard equation, we get $a = 2,b = 2$,

$h = \frac{5}{2},g = 2,f = \frac{5}{2}$ and $c = \lambda$.

We also know that the condition for a pair of straight lines is $abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0$.

Therefore, $4\lambda + 25 - \frac{25}{2} - 8 - \frac{25}{4}\lambda = 0$ or $\frac{- 9\lambda}{4} + \frac{9}{2} = 0$ or

$\lambda = 2$

Substituting value of λ in equation (i), we get

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