The point (4, 1) undergoes the following transformation successively.
(i) reflection about the line y = x
(ii) translation through a distance 2 units along the positive direction of x - axis
(iii) rotation through an angle π4 about the origin in the anticlockwise direction.
(iv) reflection aout x = 0
The final position of the given point is
(1√2, 7√2)
Let B, C, D, E be the positions of the given point A(4, 1) after the transformation (i), (ii), (iii) and (iv) successively.
The coordinates of B are (1, 4) and that of C are (1 + 2, 4 + 0), i.e., (3, 4). Now if OC makes an angle θ with x - axis, OD makes and an angle θ+π4 with x - axis. If (h, k) denote the coordinates of D, then
h=OD cos (θ+45∘), k=sin (θ+45∘)
and OD = OC = 5, sin θ=45, cos θ=35
⇒ h=(5√2) (cos θ−sin θ)=−1√2k=(5√2)(cos θ+sin θ)=7√2
coordinates of D are (−1√2, 7√2) and its reflection about x = 0 in (1√2, 7√2)