The equation of the line segment AB is y = x, If A and B lie on same side of the line mirror 2x – y =1, then image of AB has the equation

Given lines x – y = 0 –(1)  and  2x – y – 1 = 0 –(2)

Let P = point of intersection of (1) & (2)

$\begin{array}{l}xy 1\\ \begin{array}{cccc}-1& 0& 1& -1\\ -1& -1& 2& -1\end{array}\\ \Rightarrow \frac{x}{1-0}=\frac{y}{0+1}=\frac{1}{-1+2}\\ \Rightarrow \frac{x}{1}=\frac{y}{1}=1\\ \therefore p\left(x,y\right)=\left(1,1\right)\end{array}$

Let m be the slope of required line

Required Equation   y – 1 = m ( x – 1 )—(3)

Angle b/w  (1) & (2) = Angle b/w (2) & (3)

 $\begin{array}{l}\Rightarrow \left|\frac{1-2}{1+1\left(2\right)}\right|=\left|\frac{m-2}{1+2m}\right|\\ \Rightarrow \left|\frac{m-2}{1+2m}\right|=\left|\frac{1}{3}\right|\\ \left|3m-6\right|=\left|2m+1\right|\\ \therefore 3m-6=\pm \left(2m+1\right)\\ \therefore 3m-6=2m+1,3m-6=-2m-1\\ \Rightarrow m=7,5m=5\Rightarrow m=1\\ Ifm=7from\left(3\right)y-1=7\left(x-1\right)\Rightarrow y-1=7x-7\Rightarrow 7x-y=6\end{array}$