The point (4, I ) undergoes the following three transformations successively.

i . Reflection about the line y: x.

ii. Translation through a distance 2 units along the positive direction of the x-axis. 

iii. Rotation through an angle $\frac{\pi }{4}$ about the origin in the counterclockwise direction. 

Then the final position of the point is given by the co ordinates 

 

 

 Reflection about the line y = x changes the point (4, l) to (1, 4). 

On the translation of (1, 4) through a distance of 2 units along the positive direction of the x-axis, the point becomes (1 + 2. 4), i.e., (3,4). 

On rotation about the origin through an angle $\frac{\mathrm{\pi }}{4}$

point P takes the position P' such that $OP=O{P}^{\mathrm{\prime }}.\text{ Also, }OP=5$

$=O{P}^{\mathrm{\prime }}\text{ and }\cos \theta =3/5,\sin \theta =4/5\text{ . Now, }$

$\begin{array}{c}x=O{P}^{\mathrm{\prime }}\cos \left(\frac{\pi }{4}+\theta \right)\\ =5\left(\cos \frac{\pi }{4}\cos \theta -\sin \theta \right)\\ =5\left(\frac{3}{5\sqrt{2}}-\frac{4}{5\sqrt{2}}\right)\\ =-\frac{1}{\sqrt{2}}\\ y=O{P}^{\mathrm{\prime }}\sin \left(\frac{\pi }{4}+\theta \right)\\ =5\left(\sin \frac{\pi }{4}\cos \theta +\cos \frac{\pi }{4}\sin \theta \right)\\ =5\left(\frac{3}{5\sqrt{2}}+\frac{4}{5\sqrt{2}}\right)=\frac{7}{\sqrt{2}}\\ P^{\mathrm{\prime }}\equiv \left(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)\end{array}$