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

Question

Accepted Answer

To determine the image of the point (3, 8) in the line x + 3y = 7, we need to find the reflection of this point across the given line. The process involves the following steps:

1. **Equation of the Line**: The line is given by x + 3y = 7. This can be rewritten in the slope-intercept form as y = -(1/3)x + 7/3. The slope of this line is -1/3.

2. **Perpendicular Slope**: The slope of the line perpendicular to x + 3y = 7 is the negative reciprocal of -1/3, which is 3.

3. **Equation of the Perpendicular Line**: The line perpendicular to x + 3y = 7 and passing through the point (3, 8) can be written as:
   $$
   y - 8 = 3(x - 3)
   $$
   Simplifying, we get:
   $$
   y = 3x - 9 + 8
   $$
   $$
   y = 3x - 1
   $$

4. **Intersection Point**: To find the intersection of the line x + 3y = 7 and the perpendicular line y = 3x - 1, we solve the system of equations:
   $$
   x + 3(3x - 1) = 7
   $$
   $$
   x + 9x - 3 = 7
   $$
   $$
   10x = 10
   $$
   $$
   x = 1
   $$
   Substituting x = 1 into y = 3x - 1:
   $$
   y = 3(1) - 1 = 2
   $$
   So, the intersection point is (1, 2).

5. **Reflection Point**: The reflection of (3, 8) across the point (1, 2) can be found using the midpoint formula. If (x', y') is the reflection point, then:
   $$
   \frac{3 + x'}{2} = 1 \quad \text{and} \quad \frac{8 + y'}{2} = 2
   $$
   Solving these equations:
   $$
   3 + x' = 2 \times 1 \quad \Rightarrow \quad x' = 2 - 3 = -1
   $$
   $$
   8 + y' = 2 \times 2 \quad \Rightarrow \quad y' = 4 - 8 = -4
   $$
   Thus, the reflection point is (-1, -4).

Therefore, the image of the point (3, 8) in the line x + 3y = 7 is (-1, -4), which corresponds to Option 4.

**Explanation for Incorrect Options**:
- **Option 1 (1, 4)**: This point does not satisfy the conditions derived from the reflection process.
- **Option 2 (4, 1)**: This point is not equidistant from the line x + 3y = 7 as required for a reflection.
- **Option 3 (-4, -1)**: This point does not lie on the perpendicular bisector of the segment joining (3, 8) and its reflection.

Thus, Option 4 is the correct answer, as it accurately represents the reflection of the point (3, 8) across the line x + 3y = 7.