Question

Find the equations of the asymptotes of the hyperbola $3x^2+10xy+8y^2+14x+22y+7=0$

Answer 1

Given, $3x^2+10xy+8y^2+14x+22y+7=0$

standard form of hyperbola is

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ Asymptotes:-

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$

So,

$=3x^2+10xy+8y^2+14x+22y+10=0$

This is pair of straight lines.

$a=3,b=8,g=7,b=11,n=5,c=k$

$\left|\begin{array}{ccc}a& h& g\\ h& b& b\\ g& b& c\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}3& 5& 7\\ 5& 8& 11\\ 7& 11& k\end{array}\right|=0$

$\Rightarrow 3(8k-121)-5(5k-77)+7(55-56)=0$

$\Rightarrow 24k-363-25k+385-7=0$

$\Rightarrow -k-370+385=0$

$\therefore k=15$

Asymptotes are :-

$3x^2+10xy8y^2+14x+22y+15=0$