The line AB whose equation is x – y = 2 cuts the x-axis at A, and B is (4, 2). The line segment AB is rotated about A through an angle $\frac{\pi }{4}in$anticlockwise sense. Then new position B is

Equation of AB is x-y-2=0 and b=(4, 2)

Slope= -1/-1 = 1

AB makes an angle 45⁰ with x-axis

AB meets x-axis at A (2, 0)

Let  $\theta =$ Angle made by AC with x-axis

$\begin{array}{l}=\frac{\pi }{4}+\frac{\pi }{4}=\frac{\pi }{2}\\ LetA\left(2,0\right)=\left(x_1{y}_1\right)B\left(4,2\right)\Rightarrow r=AB=\sqrt{2^2+2^2}=2\sqrt{2}\\ \therefore C=\left(x_1+r\cos \theta ,y_1+r\sin \theta \right)\\ =\left(2+2\sqrt{2}\cos {90}^0,0+2\sqrt{2}\sin {90}^0\right)\\ =\left(2+0,0+2\sqrt{2}\right)\\ =\left(2,2\sqrt{2}\right)\end{array}$