The point (4,1) undergoes the following three successive transformations ,
reflection about the line `y = x-1`
translation through a distance 1 unit along the positive direction
rotation thrpough an angle `pi/4` about the origin in the anti - clockwise direction
Then the coordinates of the final point are ,

Options

A. `(4,3)`
B. `(7/2,7/2)`
C. `(0,3sqrt(2))`
D. `(3,4)`

Answer

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

### Step 1: Reflection about the line \( y = x - 1 \)

To reflect a point \((x_1, y_1)\) about the line \(y = mx + c\), we can use the formula for the reflection of a point across a line. The line \(y = x - 1\) can be rewritten in the standard form \(x - y + 1 = 0\).

Using the reflection formula, the coordinates of the reflected point \((x', y')\) are given by:

\[ x' = \frac{x_1(1-m^2) + 2my_1 - 2mc}{1+m^2} \]
\[ y' = \frac{y_1(m^2-1) + 2mx_1 + 2c}{1+m^2} \]

Here, \(m = 1\) (the slope of the line), and \(c = 1\).

Substituting \(x_1 = 4\) and \(y_1 = 1\):

\[ x' = \frac{4(1-1^2) + 2(1)(1) - 2(1)}{1+1^2} = \frac{0 + 2 - 2}{2} = 0 \]

\[ y' = \frac{1(1^2-1) + 2(1)(4) + 2(1)}{1+1^2} = \frac{0 + 8 + 2}{2} = 5 \]

Thus, the coordinates after reflection are \((0, 5)\).

### Step 2: Translation through a distance of 1 unit along the positive x-direction

To translate the point \((0, 5)\) by 1 unit along the positive x-direction, we simply add 1 to the x-coordinate:

\[ (0 + 1, 5) = (1, 5) \]

### Step 3: Rotation through an angle of \(\frac{\pi}{4}\) about the origin in the anti-clockwise direction

To rotate a point \((x, y)\) about the origin by an angle \(\theta\), we use the following transformation:

\[ x' = x \cos(\theta) - y \sin(\theta) \]
\[ y' = x \sin(\theta) + y \cos(\theta) \]

For \(\theta = \frac{\pi}{4}\), we have \(\cos(\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\).

Substituting \((x, y) = (1, 5)\):

\[ x' = 1 \cdot \frac{1}{\sqrt{2}} - 5 \cdot \frac{1}{\sqrt{2}} = \frac{1 - 5}{\sqrt{2}} = \frac{-4}{\sqrt{2}} = -2\sqrt{2} \]

\[ y' = 1 \cdot \frac{1}{\sqrt{2}} + 5 \cdot \frac{1}{\sqrt{2}} = \frac{1 + 5}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2} \]

Thus, the final coordinates after all transformations are \((-2\sqrt{2}, 3\sqrt{2})\).

### Final Answer:
The coordinates of the final point are \((-2\sqrt{2}, 3\sqrt{2})\).

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