The point (4,1) undergoes the following three transformations successively
(i) Reflection about the line y=x
(ii) Transformation through a distance 2 units along the positive direction of x-axis
(iii) Rotation through angle `pi//4` about the origin in the anticlockwise direction. The final position of the point is given by the coordinates

Options

A. `(1/(sqrt(2)),7/(sqrt(2)))`
B. `(-2,7sqrt(2))`
C. `(-1/(sqrt(2)),7/(sqrt(2)))`
D. `(sqrt(2),7sqrt(2))`

Answer

To solve the problem, we will perform each transformation step by step on the point (4, 1).

### Step 1: Reflection about the line y = x
When we reflect a point (x, y) about the line y = x, the coordinates are swapped.

For the point (4, 1):
- After reflection: (1, 4)

### Step 2: Transformation through a distance of 2 units along the positive direction of the x-axis
To move a point (x, y) along the positive x-axis by a distance of d, we add d to the x-coordinate.

For the point (1, 4):
- After moving 2 units along the x-axis: (1 + 2, 4) = (3, 4)

### Step 3: Rotation through an angle of π/4 about the origin in the anticlockwise direction
To rotate a point (x, y) about the origin by an angle θ, we use the rotation formulas:
- New x-coordinate = x * cos(θ) - y * sin(θ)
- New y-coordinate = x * sin(θ) + y * cos(θ)

Here, θ = π/4, and we know:
- cos(π/4) = sin(π/4) = √2/2

For the point (3, 4):
- New x-coordinate = 3 * (√2/2) - 4 * (√2/2) = (3 - 4) * (√2/2) = -1 * (√2/2) = -√2/2
- New y-coordinate = 3 * (√2/2) + 4 * (√2/2) = (3 + 4) * (√2/2) = 7 * (√2/2)

Thus, after rotation, the new coordinates are:
- Final position: (-√2/2, 7√2/2)

### Final Answer
The final position of the point after all transformations is given by the coordinates:
(-√2/2, 7√2/2)

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