Question

The value of $\lambda$ for which the equation$2x^2+7xy+3y^2+8x+14y+\lambda =0$ represents a pair of straight lines is:

• A. $1$
• B. $2$
• C. $4$
• D. $8$

Answer 1

The correct option is D $8$

We know, $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a pair of straight lines if
$\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}2& \frac{7}{2}& 4\\ \frac{7}{2}& 3& 7\\ 4& 7& \lambda \end{array}\right|=0$
$\Rightarrow 2(3\lambda -49)-\frac{7}{2}(\frac{7}{2}\lambda -28)+4(\frac{49}{2}-12)=0$
$\Rightarrow 6\lambda -98-\frac{49\lambda }{4}+98+98-48=0\Rightarrow \frac{49\lambda }{4}-6\lambda =50\Rightarrow \frac{25\lambda }{4}=50\therefore \lambda =\frac{200}{25}=8$