Question

The value of $\lambda$, for which the equation $6x^2+11xy-10y^2+x+3+y+\lambda =0$ represents a pair of straight lines, is

• A. $-15$
• B. $0$
• C. $2$
• D. None of these

Answer 1

The correct option is D None of these

Compare the equation $6x^2+11xy-10y^2+x+3+y+\lambda =0$ with $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$
$\therefore a=6,b=-10,h=\frac{11}{2},g=\frac{1}{2},f=\frac{1}{2},c=3+\lambda$
Equation represent the pair of straight lines if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$
$\Rightarrow 6\times (-10)\times (3+\lambda )+2\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{11}{2}-6\cdot \frac{1}{4}-(-10)\times \frac{1}{4}-(3+\lambda )\cdot \frac{121}{4}=0$
$\Rightarrow \lambda =-\frac{713}{241}$