In coordinate geometry, what is the reflection of the point (4, -3) in the horizontal line given by the equation y = -2?

Introduction / Context:
Reflection of points in lines is a basic topic in coordinate geometry. It often appears in school level mathematics and in competitive exams that test conceptual clarity. In this problem, we are asked to find the reflection of a point in a horizontal line, which is a very common situation. Understanding how distances behave with respect to such lines makes this question straightforward.

Given Data / Assumptions:

Original point is P with coordinates (4, -3).
The line of reflection is the horizontal line y = -2.
Reflection is taken in the usual geometric sense, meaning the line is the perpendicular bisector of the segment joining the point and its image.
The coordinate axes and distance measurements follow standard Cartesian conventions.

Concept / Approach:
For reflection across a horizontal line of the form y = k, the x coordinate of a point does not change, because horizontal lines are parallel to the x axis. Only the y coordinate changes. The reflected point lies vertically above or below the original point such that the reflecting line y = k is the midpoint in the vertical direction. In other words, the vertical distance from the original point to the line equals the vertical distance from the line to the reflected point, but on the opposite side.

Step-by-Step Solution:
Step 1: Identify the given line of reflection as y = -2. This is a horizontal line one unit above the line y = -3 and one unit below the line y = -1. Step 2: Write the coordinates of the original point P as (4, -3). Here x = 4 and y = -3. Step 3: Compute the vertical distance from P to the line y = -2. Distance in the y direction is absolute value of (-3 - (-2)) which equals 1. Step 4: To find the reflection, move the same distance of 1 unit on the other side of the line. Since the line is at y = -2, moving 1 unit above gives y = -1. Step 5: Keep the x coordinate unchanged because reflection across a horizontal line does not affect x. Therefore, the reflected point has x = 4 and y = -1, so the image point is (4, -1).

Verification / Alternative check:
We can verify by checking that the line y = -2 is the midpoint between the original and the reflected point along the vertical direction. The original y value is -3 and the reflected y value is -1. The midpoint of these two y values is ((-3) + (-1)) / 2 which equals -4 / 2 or -2. This matches the equation of the reflecting line y = -2. Also, the x coordinates of both points are equal, confirming that the points lie on a vertical line, which is perpendicular to y = -2. This confirms that (4, -1) is indeed the correct reflection.

Why Other Options Are Wrong:
(4, 1): This point is much farther vertically from y = -2 and does not make y = -2 the midpoint between -3 and 1, so it is not a correct reflection.
(-4, 1): Both x and y coordinates change here. Reflection in a horizontal line should not change the x coordinate, so this is incorrect.
(-4, -1): This point changes x from 4 to -4, which does not happen for reflection across a horizontal line. Therefore, it is not the correct image.

Common Pitfalls:
A common mistake is to change both x and y coordinates when reflecting across a line, or to reflect the point across the x axis (y = 0) instead of the specified line y = -2. Another frequent error is to miscalculate the vertical distance and move two units instead of one unit. To avoid these issues, always identify the form of the reflecting line, keep the coordinate parallel to the line unchanged, and apply equal distances on opposite sides of the line for the orthogonal coordinate.

Final Answer:
The reflection of the point (4, -3) in the line y = -2 is (4, -1).