$\text{ Line }y=2x-b\text{ cuts the parabola }y=x^2-4x\text{ at points }A\text{ and }B\text{ . The value of }b\text{ for which the }\mathrm{\angle }AOB\text{ is a right angle is}$

$\begin{array}{r}\text{ Line }y=2x-b\\ \Rightarrow 1=\frac{2x-y}{b}\end{array}$

$\begin{array}{r}\text{ Homogenizing parabola with line }\\ x^2-4x\left(\frac{2x-y}{b}\right)-y\left(\frac{2x-y}{b}\right)=0\end{array}$

$\begin{array}{l}\text{ Since }\mathrm{\angle }AOB={90}^{\circ }\\ \therefore \text{ coefficient of }{x}^2+\text{ coefficient of }{y}^2=0\end{array}$

$\begin{array}{l}\Rightarrow 1-\frac{8}{b}+\frac{1}{b}=0\\ \Rightarrow b=7\end{array}$