Question

Find value of $\lambda$ for which line $y=x+1,y=\lambda x+2$ and $y=({\lambda }^2+\lambda -1)x+3$ are con-current.

Answer 1

$\begin{array}{c}3linesareconcurrentifthepointofinstructionof\\ 2linesliesonthethirdline\\ y=x+1\to \left(1\right)\\ y=\lambda x+2\to \left(2\right)\\ y=\left({\lambda }^2+\lambda -1\right)x+3\to \left(3\right)\\ solvingequation\left(1\right)and\left(2\right)\\ x=\frac{1}{1-\lambda }y=\frac{2-\lambda }{1-\lambda }\\ \left(\frac{1}{1-\lambda },\frac{2-\lambda }{1-\lambda }\right)mustsatisfyequation\left(3\right)\\ \Rightarrow \frac{2-\lambda }{1-\lambda }=\left({\lambda }^2+\lambda -1\right)\times \frac{1}{1-\lambda }+3\\ \Rightarrow {\lambda }^2-\lambda =0\\ \Rightarrow \lambda \left(\lambda -1\right)=0\Rightarrow \lambda =0,1\\ \lambda ca{n}^{\prime }tbe1aspointofintersectionhas1-\lambda asi{t}^{\prime }sdenominator\end{array}$