The image of the point (3,8) in the line x + 3y = 7 is

Options

A. (4, 7)
B. (2, 3)
C. (- 1, - 4)
D. None of these

Answer

To find the image of the point (3,8) in the line x + 3y = 7, we can follow these steps:

### Step 1: Identify the line equation
The given line equation is:
\[ x + 3y = 7 \]

### Step 2: Find the slope of the line
To find the slope of the line, we can rearrange the equation into the slope-intercept form (y = mx + b):
\[ 3y = -x + 7 \\ y = -\frac{1}{3}x + \frac{7}{3} \]
The slope (m) of the line is -1/3.

### Step 3: Find the perpendicular slope
The slope of the line perpendicular to this line will be the negative reciprocal of -1/3, which is:
\[ m_{\text{perpendicular}} = 3 \]

### Step 4: Write the equation of the perpendicular line
Using the point (3,8) and the slope of the perpendicular line, we can use the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \\ y - 8 = 3(x - 3) \]
Expanding this gives:
\[ y - 8 = 3x - 9 \\ y = 3x - 1 \]

### Step 5: Find the intersection of the two lines
Now, we need to find the intersection of the lines \( y = 3x - 1 \) and \( x + 3y = 7 \). Substitute \( y \) from the second equation into the first:
\[ x + 3(3x - 1) = 7 \\ x + 9x - 3 = 7 \\ 10x - 3 = 7 \\ 10x = 10 \\ x = 1 \]

### Step 6: Substitute x back to find y
Now substitute \( x = 1 \) back into the equation of the perpendicular line to find \( y \):
\[ y = 3(1) - 1 = 2 \]

### Step 7: Find the image point
The intersection point is (1, 2). The image of the point (3, 8) across the line is found by reflecting it over this intersection point. To find the image point \( (x', y') \), we can use the midpoint formula:
\[ \left( \frac{x + x'}{2}, \frac{y + y'}{2} \right) = (1, 2) \]
This gives us the equations:
\[ \frac{3 + x'}{2} = 1 \quad \text{and} \quad \frac{8 + y'}{2} = 2 \]

Solving these:
1. For \( x' \):
\[ 3 + x' = 2 \\ x' = 2 - 3 \\ x' = -1 \]

2. For \( y' \):
\[ 8 + y' = 4 \\ y' = 4 - 8 \\ y' = -4 \]

### Final Answer
The image of the point (3, 8) in the line \( x + 3y = 7 \) is:
\[ (-1, -4) \]