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The point (4, 1) undergoes the following...

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

(2,3)

B

`(2,3sqrt2)`

C

(0,3`sqrt2`)

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Let Q (`x_(1),y_(1)`) be the reflection of the point P(4,1)in the line mirror y=x-1. Then, the coordinates of Q are given by
`(x_(1)-4)/1=(y_(1)-1)/(-1)=(-2(4-1-1))/({1^(2)+(-1)^(2)})`
`rArr(x_(1)-4)/1=(y_(1)-1)/(-1)=-2rArrx_(1)=2,y_(1)=3`
Thus, the coordinates of Q are (2,3)
Now, Q is translated parallel to x-axis in the positive direction by 1 unit. So, the coordiantes of R aer (1+2,3) or (3,3) .
Suppose OR makes an angle `theta` with OX.Then,
`tan theta=(3)/(3)=1 implies theta=(pi)/(4)`
It is given that OR is rotated through `(pi)/(4)` in anticlockwise sense to coincide with OS. Therefore, OS makes a right angle with OX and OS =OR`=3sqrt(2)` .
Thus, the coordiantes of S are `(0,3sqrt(2))`.
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