What are the co ordinates of the reflection of the point (−4, 3) in the vertical line x = −2 on the Cartesian plane?

Introduction / Context:
This coordinate geometry question asks you to find the reflection of a point across a vertical line. Reflection is a basic transformation where every point is mapped to another point such that the reflecting line is the perpendicular bisector of the segment joining the original point and its image. For a vertical line of the form x = constant, only the x co ordinate changes, while the y co ordinate remains the same. Understanding this helps solve reflection problems quickly in aptitude exams.

Given Data / Assumptions:
- Original point P has co ordinates (−4, 3). - The reflecting line is the vertical line x = −2. - We must find the co ordinates of the reflected point P prime. - The reflection is defined so that line x = −2 is midway between the x co ordinates of P and P prime.

Concept / Approach:
For reflection across a vertical line x = a, the y co ordinate remains unchanged, because vertical reflection only moves the point horizontally. The x co ordinate transforms according to the idea that the reflecting line is the midpoint between the original and image x values. Algebraically, if the original x is x1 and the new x is x2, then the average (x1 + x2) / 2 equals a. We can use this formula directly, or use the shortcut x2 = 2a − x1. The y co ordinate stays the same as that of the original point.

Step-by-Step Solution:
Step 1: Identify the reflecting line x = −2, so here a = −2. Step 2: Original point P has co ordinates (x1, y1) = (−4, 3). Step 3: For reflection across x = a, the new x co ordinate is x2 = 2a − x1. Step 4: Substitute a = −2 and x1 = −4 to get x2 = 2 * (−2) − (−4) = −4 + 4 = 0. Step 5: The y co ordinate remains unchanged, so y2 = y1 = 3. Step 6: Therefore the reflected point P prime has co ordinates (0, 3).

Verification / Alternative check:
To verify, check that the reflecting line x = −2 is the midpoint in the horizontal direction. Compute the midpoint of the segment joining P(−4, 3) and P prime(0, 3). The average of the x co ordinates is (−4 + 0) / 2 = −2, which is exactly the x value of the reflecting line. The y co ordinates are both 3, so the reflecting line is indeed perpendicular and acts as the perpendicular bisector. This confirms that (0, 3) is the correct reflection.

Why Other Options Are Wrong:
Option (−4, −7) changes the y co ordinate and keeps x the same, which corresponds to reflection across a horizontal line, not across x = −2. Option (4, 3) would place the point 6 units to the right of the reflecting line, while the original point is only 2 units to the left, so the line would not be the midpoint. Option (−4, 7) again changes only the y co ordinate, which does not match reflection in a vertical line.

Common Pitfalls:
Students sometimes wrongly change the y co ordinate instead of the x co ordinate when reflecting across a vertical line. Another mistake is to reflect across the y axis (line x = 0) instead of the given line x = −2. Always note that for x = a, distances to the left and right of the line must be equal in magnitude for a reflection. The shortcut formula x2 = 2a − x1 is very effective and helps avoid sign errors.

Final Answer:
The reflection of the point (−4, 3) in the line x = −2 is (0, 3).