The point (4,1) undergoes the following two successive transformations
(i) Reflection about the line `y=x`
(ii) Translation through a distance of 2 units along the positive x-axis.
The coordinates of the new point are

Options

A. `(-(1)/(sqrt(2)),(7)/(sqrt(2)))`
B. `(-(1)/(2),(7)/(sqrt(2)))`
C. `((7)/(sqrt(2)),(1)/(2))`
D. none of these

Answer

B

Explanation

Let Q be the reflection of P(4,1) about the line y=x . Then, the coordinates of Q are (1,4) the second transformation the point Q is translated through distance of 2 units along the positive direction of x-axis . Let the new position of Q be R. Then, the coordinates of R are (1+2,4) i.e. (3,4).
Suppose OR makes an angle `theta` with `OX`, Then,
`tan theta=(4)/(3), sin theta=(4)/(5) and cos theta =(3)/(5)`
In the third transformation OR is rotated through `pi//4` in anticlockwise direction of coincide with OS. We have to find the co-ordinates of S.
Clearly , OR=OS .
`therefore OS=sqrt(3^(2)+4^(2))`=5
OS makes angle `(theta+pi//4)` with OX. So, the coordinates of S are
`(OS cos (theta + pi//4), OS sin (theta+pi//4))`
or , `(5((costheta-sin theta)/(sqrt(2))),5((cos theta + sin theta)/(sqrt(2))))" or ", (-(1)/(2),(7)/(sqrt(2)))`
Hence , the coordiantes of P in the final position are `(-1/2,7/sqrt(2)`.