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.

Explanation

Let `Q(x_(1),y_(1))` be the reflection of `P(4,1)` about the line `y=x.`
Then `x_(1)=1,y_(1)=4.`
i.e., coordinates of Q are (1,4)
For euestion Q move 2 units along the positive direction of x-axis
`therefore` coordinates of R is `(1+2,4)` i.e.,`R(3,4)`
Let OR make an angle `theta` with positive direction of x-axis, then `tan theta=(4)/(3)and OR=5`
`impliessin theta=4/5and costheta=3/5`
After rotation of `(pi)/(4)` in anticlockwise direction about origin, let S be the final position of R.
Then `OS=OR=5and OS` make an angle `(pi)/(4)+theta` with x-axis
Hence coordinates of A are `(OS cos(theta+(pi)/(4)),OSsin(theta+(pi)/(4)))`
`S(5((1)/(sqrt2)costheta-(2)/(sqrt2)sintheta),5(sin thetaxx(1)/(sqrt2)+(1)/(sqrt2)cos theta))`
`=S(5((3)/(5sqrt2)-(4)/(5sqrt2)),5((3)/(5sqrt2)+(4)/(5sqrt2)))`
`i.e.,R((-1)/(sqrt2,),(7)/(sqrt2))`