Find the new coordinates of $(2,3)$ if following transformations take place: (i) Origin is shifted to $(1,1)$ (ii) Axes are rotated by an angle $45^{\circ}$ in anticlockwise sense (iii) Origin is shifted to $(1,1)$ and then axes are rotated by angle $45^{\circ}$ in clockwise sense (iv) Coordinate axes become $x+y+1=0$ and $x-y+2=0$
Here is what I have tried:
At first step, coordinates become $(1,2)$, then using $X=x\cos\theta+y\sin\theta, Y=y\cos\theta-x\sin\theta$, we get at the second step $(3/\sqrt{2}, 1/\sqrt{2})$. Now again translating origin to $(1,1)$, and applying rotation, this time $45^{\circ}$ clockwise we get $(1,2-\sqrt{2})$. Finally, changing axes to given lines, we get using the following formula, the point as $(2\sqrt{2}-1,1+1\sqrt{2})$
$$X=\frac{lx+my+n}{\sqrt{l^2+m^2}}\\ Y=\frac{mx-ly+n^{'}}{\sqrt{l^2+m^2}}$$
Turns out the actual answer given is $(1/\sqrt{2}, 6/\sqrt{2})$. Can anybody spot the mistake? Thanks.