Question

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

Answer 1

$2x^2+5xy+2y^2+4x+5y=0$ $\to$ equation of hyperbola

Equation of Asymptoms & Hyperbola differ by a constant

So, lets take equation of asymptotes as

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

(pair of lines)

$\Rightarrow \Delta =0=abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

Here, $a=2,h=\frac{5}{2},c=c,b=z,g=2,f=\frac{5}{2}$

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$4c+25-\frac{25}{2}-8-c\left(\frac{25}{4}\right)=0$

$4c+\frac{25}{2}-8-\frac{25c}{4}=0$

$\frac{9}{2}+\frac{(16x-25c)}{4}=0$

$\frac{9}{2}=\frac{9c}{4}$

$\Rightarrow c=2$.