The position of reflection of the point (4,1) about the line y=x -1 is

Options

A. 1,2
B. 3,4
C. `-1,0`
D. 2,3

Explanation

To find the position of the reflection of the point \( P(4, 1) \) about the line \( y = x - 1 \), we can follow these steps:

### Step 1: Identify the line and the point
The line of reflection is given by the equation \( y = x - 1 \). The point to be reflected is \( P(4, 1) \).

### Step 2: Find the coordinates of point R on the line
Let the coordinates of point \( R \) (the foot of the perpendicular from \( P \) to the line) be \( (h, k) \). Since \( R \) lies on the line \( y = x - 1 \), we can express \( k \) in terms of \( h \):
\[ k = h - 1 \]

### Step 3: Calculate the slope of line PR
The slope of line \( PR \) can be calculated using the formula for the slope between two points:
\[ \text{slope of } PR = \frac{k - 1}{h - 4} \]
Substituting \( k = h - 1 \):
\[ \text{slope of } PR = \frac{(h - 1) - 1}{h - 4} = \frac{h - 2}{h - 4} \]

### Step 4: Determine the slope of the line of reflection
The slope of the line \( y = x - 1 \) is \( 1 \).

### Step 5: Use the property of perpendicular lines
Since \( PR \) is perpendicular to the line of reflection, the product of their slopes must equal \(-1\):
\[ \frac{h - 2}{h - 4} \cdot 1 = -1 \]
This simplifies to:
\[ h - 2 = - (h - 4) \]
\[ h - 2 = -h + 4 \]
Adding \( h \) to both sides:
\[ 2h - 2 = 4 \]
Adding \( 2 \) to both sides:
\[ 2h = 6 \]
Dividing by \( 2 \):
\[ h = 3 \]

### Step 6: Find the value of k
Substituting \( h = 3 \) back into the equation for \( k \):
\[ k = h - 1 = 3 - 1 = 2 \]
Thus, the coordinates of point \( R \) are \( (3, 2) \).

### Step 7: Use the midpoint formula to find Q
Since \( R \) is the midpoint of \( P \) and \( Q \), we can use the midpoint formula:
\[ R = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Let \( Q = (a, b) \). Then:
\[ 3 = \frac{4 + a}{2} \quad \text{and} \quad 2 = \frac{1 + b}{2} \]

### Step 8: Solve for a and b
From the first equation:
\[ 3 \cdot 2 = 4 + a \implies 6 = 4 + a \implies a = 2 \]
From the second equation:
\[ 2 \cdot 2 = 1 + b \implies 4 = 1 + b \implies b = 3 \]

### Conclusion
The coordinates of the reflection point \( Q \) are \( (2, 3) \).

Thus, the position of the reflection of the point \( (4, 1) \) about the line \( y = x - 1 \) is \( \boxed{(2, 3)} \).