Question

The equation of the asymptotes of the hyperbola $2x^2-5xy-3y^2-5x-3y-21=0$ differ by $\lambda$ then $\lambda =?$

• A. $\frac{18}{49}$
• B. $\frac{49}{18}$
• C. $\frac{-18}{49}$
• D. None of these

Answer 1

The correct option is C $\frac{-18}{49}$

Since the equation is the pair of asymptotes of the hyperbola differ by the hyperbola by a constant only
$\therefore$ equation of pair of asymptotes be
$2x^2-5xy-3y^2-5x-3y+\lambda =0$ which represents itself a pair of straight lines, the second degree equation
$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represent a pair of straight lines if
$abc-2fgh-a{f}^2-b{g}^2-c{h}^2=0$
$\therefore -6\lambda -\frac{75}{4}-\frac{9}{2}+\frac{75}{4}-\frac{25}{4}\lambda =0$
$\Rightarrow -49\lambda -75-18+75=0\Rightarrow \lambda =\frac{-18}{49}$
Hence, option 'C' is correct.