Question

The value of $ \lambda $ , for which the equation $ x ^2 - y^ 2 - x + \lambda y - 2 = 0 $ represents a pair of straight lines, are :

• A. $ -3, 1 $
• B. $ -1, 1 $
• C. $ -3, 3 $
• D. $ 3, 1 $

Answer 1

Answer: (C)

We have, $x^2 - y^2 -x + \lambda y -2 = 0 \quad ...(i)$
We know that,
$ax^2 + 2 hxy + by^2 + 2 gx + 2fy + c = 0\quad ...(ii)$
Now comparing Eqs. $(i)$ and $(ii)$, we get
$a = 1, b = -1, c =-2, h = 0, g =-1/2$,
$f = λ / 2$
$∴$ Eq. $(ii)$ represents a pair of straight line, if
$ abc + 2fgh - af^2 - bg^2 - ch^2 = 0$
$\therefore 2 +0 - \frac{\lambda^{2}}{4} + \frac{1}{4} = 0 $
$\Rightarrow \frac{\lambda^{2}}{4} = \frac{9}{4}$
$ \Rightarrow \lambda = \pm 3$