Question

If the lines $L_1:{\lambda }^{2}x-y-1=0$ $L_2:x-{\lambda }^{2}y+1=0$ $L_3:x+y-{\lambda }^2=0$ pass through the same point the value(s) of $\lambda$ equals

• A. $1$
• B. $\sqrt{2}$
• C. $2$
• D. $0$

Answer 1

Answer: (B)

$\sqrt{2}$

The given equations passes through same point.So, they are concurrent lines
$\Rightarrow \left|\begin{array}{ccc}{\lambda }^2& -1& -1\\ 1& -{\lambda }^2& 1\\ 1& 1& -{\lambda }^2\end{array}\right|=0$
$\Rightarrow {\lambda }^6-3{\lambda }^2-2=0$

By doing synthetic division we get,

$({\lambda }^4-2{\lambda }^2+1)({\lambda }^2-2)=0$

$({\lambda }^2-1{)}^2({\lambda }^2-2)=0$

$\lambda =\pm \sqrt{2}or\lambda =\pm 1$

But here $\lambda =\pm 1doesnotsatisfies,hence\lambda =\sqrt{2}$
Option B satisfies above equation