The image of the point (1, 3) in the line `x + y-6 = 0` is

Options

A. (3,5)
B. (5,3)
C. (1,-3)
D. (-1,3)

Answer

To find the image of the point (1, 3) with respect to the line \( x + y - 6 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients from the line equation
The line equation can be rewritten in the standard form \( ax + by + c = 0 \):
\[ x + y - 6 = 0 \implies 1x + 1y - 6 = 0 \]
Thus, we have:
- \( a = 1 \)
- \( b = 1 \)
- \( c = -6 \)

### Step 2: Identify the point coordinates
The given point is \( P(1, 3) \), where:
- \( x_1 = 1 \)
- \( y_1 = 3 \)

### Step 3: Use the formula for the image of a point with respect to a line
The formula for finding the image \( P' (x, y) \) of a point \( P(x_1, y_1) \) with respect to the line \( ax + by + c = 0 \) is given by:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{-2(ax_1 + by_1 + c)}{a^2 + b^2} \]

### Step 4: Calculate the right-hand side of the formula
First, calculate \( -2(ax_1 + by_1 + c) \):
\[ ax_1 + by_1 + c = 1(1) + 1(3) - 6 = 1 + 3 - 6 = -2 \]
Thus,
\[ -2(ax_1 + by_1 + c) = -2(-2) = 4 \]

Next, calculate \( a^2 + b^2 \):
\[ a^2 + b^2 = 1^2 + 1^2 = 1 + 1 = 2 \]

### Step 5: Substitute values into the formula
Now, we substitute into the formula:
\[ \frac{x - 1}{1} = \frac{y - 3}{1} = \frac{4}{2} \]
This simplifies to:
\[ \frac{x - 1}{1} = \frac{y - 3}{1} = 2 \]

### Step 6: Solve for \( x \) and \( y \)
From \( \frac{x - 1}{1} = 2 \):
\[ x - 1 = 2 \implies x = 3 \]

From \( \frac{y - 3}{1} = 2 \):
\[ y - 3 = 2 \implies y = 5 \]

### Conclusion
The image of the point \( (1, 3) \) in the line \( x + y - 6 = 0 \) is \( P'(3, 5) \).

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