The point (4, 1) undergoes the following three transformations successively: (a) Reflection about the line y = x (b) Translation through a distance 2 units along the positive direction of the x-axis. (c) Rotation through an angle `pi/4` about the origin in the anti clockwise direction. The final position of the point is given by the co-ordinates.A. `((1)/(sqrt(2)),(7)/(sqrt(2)))`B. `(-sqrt(2),7sqrt(2))`C. `(-(1)/(sqrt(2)),(7)/(sqrt(2)))`D. `(sqrt(2),7sqrt(2))`

The point (4, 1) undergoes the following three transformations success

1.	The point (4, 1) undergoes the following three transformations successively: (a) Reflection about the line y = x (b) Translation through a distance 2 units along the positive direction of the x-axis. (c) Rotation through an angle `pi/4` about the origin in the anti clockwise direction. The final position of the point is given by the co-ordinates.A. `((1)/(sqrt(2)),(7)/(sqrt(2)))`B. `(-sqrt(2),7sqrt(2))`C. `(-(1)/(sqrt(2)),(7)/(sqrt(2)))`D. `(sqrt(2),7sqrt(2))`
Answer» Let `B,C,D` be the position of the point `A(4,1)` after the three operations `I,II` and `III` respectively. Then, `B` is `(1,4), C(1+2,4)` i.e. `(3,4)` . The point `D` is obtained from `C` by rotationg the coordinate axes through an angle `pi//4` in anti-clockwise direction. Therefore, the coordinates of `D` are given by `X=3cos.(pi)/(4)-4sin.(pi)/(4)=-(1)/(sqrt(2))` and `Y=3sin.(pi)/(4)+4cos.(pi)/(4)=(7)/(sqrt(2))` `:.` Coordinates of `D` are `(-(1)/(sqrt(2)),(7)/(sqrt(2)))`.

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