The image of the point (3, 5) in the line `x -y + 1 = 0`, lies on :

Options

A. `(x-2)^(2)+(y-2)^(2)=12`
B. `(x-4)^(2)+(y+2)^(2)=16`
C. `(x-4)^(2)+(y-4)^(2)=8`
D. `(x-2)^(2)+(y-4)^(2)=4`

Answer

To find the image of the point (3, 5) in the line given by the equation \(x - y + 1 = 0\), we can follow these steps:

### Step 1: Identify the line equation and point
The line equation is given as:
\[ x - y + 1 = 0 \]
This can be rewritten as:
\[ y = x + 1 \]
The point we are considering is \(P(3, 5)\).

### Step 2: Find the slope of the line
The slope of the line \(y = x + 1\) is \(1\). The slope of the perpendicular line (which will help us find the image) is the negative reciprocal of \(1\), which is \(-1\).

### Step 3: Write the equation of the perpendicular line
Using the point-slope form of the line equation, the equation of the line passing through point \(P(3, 5)\) with slope \(-1\) is:
\[ y - 5 = -1(x - 3) \]
Simplifying this, we get:
\[ y - 5 = -x + 3 \implies y = -x + 8 \]

### Step 4: Find the intersection of the two lines
Now, we need to find the intersection of the lines \(y = x + 1\) and \(y = -x + 8\). Setting them equal to each other:
\[ x + 1 = -x + 8 \]
Solving for \(x\):
\[ 2x = 7 \implies x = \frac{7}{2} \]
Substituting \(x = \frac{7}{2}\) back into one of the line equations to find \(y\):
\[ y = \frac{7}{2} + 1 = \frac{9}{2} \]
Thus, the intersection point is:
\[ \left(\frac{7}{2}, \frac{9}{2}\right) \]

### Step 5: Find the image of the point
The image \(P'\) of point \(P(3, 5)\) across the line is found by reflecting \(P\) over the intersection point. The coordinates of the image can be calculated as follows:
\[ P' = \left(2 \cdot \frac{7}{2} - 3, 2 \cdot \frac{9}{2} - 5\right) \]
Calculating the x-coordinate:
\[ 2 \cdot \frac{7}{2} - 3 = 7 - 3 = 4 \]
Calculating the y-coordinate:
\[ 2 \cdot \frac{9}{2} - 5 = 9 - 5 = 4 \]
Thus, the image point \(P'\) is:
\[ P'(4, 4) \]

### Step 6: Verify which option contains the point (4, 4)
Now we need to check which option includes the point \(P'(4, 4)\).

1. **Option 1:** Check if \(4 - 2 = 2\) and \(2^2 + 4 = 4\) (not satisfied).
2. **Option 2:** Check if \(4 - 2 = 2\) and \(2^2 + 4 = 8\) (satisfied).
3. **Option 3:** Check if \(4 + 4 = 8\) (not satisfied).
4. **Option 4:** Check if \(4 + 4 = 8\) (not satisfied).

The correct option is **Option 2**.