In coordinate geometry with reflections, a point R(a, b) is first reflected in the origin to get R₁, and then R₁ is reflected in the x-axis to become the point (-5, 1). Find the coordinates of the original point R.

Introduction / Context:
This coordinate geometry question involves successive reflections of a point in the origin and then in the x-axis. You are given the final image after the two reflections and must determine the original point. Understanding how reflections change coordinates is important for geometry, transformations, and many aptitude-style reasoning problems involving symmetry.

Given Data / Assumptions:

• Original point: R(a, b).
• Reflection of R in the origin gives R₁.
• Reflection of R₁ in the x-axis gives the point (−5, 1).
• We assume standard reflection rules in the Cartesian plane.

Concept / Approach:
Reflection in the origin transforms a point (x, y) to (−x, −y). Reflection in the x-axis transforms (x, y) to (x, −y). We apply these transformations step by step starting from R(a, b). First, we obtain R₁ by reflecting in the origin. Then we reflect R₁ in the x-axis to match the given final coordinates, and we solve for a and b. This reverse engineering process from the final point leads to the original coordinates.

Step-by-Step Solution:
1) Let the original point be R(a, b). 2) Reflect R in the origin to get R₁. Reflection in the origin changes (x, y) to (−x, −y). Thus R₁ = (−a, −b). 3) Reflect R₁ in the x-axis. Reflection in the x-axis changes (x, y) to (x, −y). So the reflection of (−a, −b) becomes (−a, b). 4) We are told this final point equals (−5, 1). Therefore, (−a, b) = (−5, 1). 5) Equate coordinates: −a = −5 and b = 1. 6) From −a = −5, we get a = 5. 7) Therefore the original point R is (a, b) = (5, 1).

Verification / Alternative check:
Check the transformations explicitly using R(5, 1). Reflecting R in the origin gives R₁ = (−5, −1). Reflecting this in the x-axis changes (−5, −1) to (−5, 1). This matches the given final point, confirming that R(5, 1) is correct. No other option, when processed through the same two reflections, produces (−5, 1).

Why Other Options Are Wrong:
Option a, (5, −1), reflected in the origin becomes (−5, 1), and reflecting that in the x-axis gives (−5, −1), which is not the given final point. Option b, (−1, 5), and option c, (1, −5), similarly produce different final points after the two reflections. Option e, (−5, 1), is the final image itself, not the original R. Only option d, (5, 1), yields (−5, 1) after the prescribed sequence of reflections.

Common Pitfalls:
A common mistake is reversing the order of reflections or mixing up the rules for each reflection. For example, some students mistakenly think reflection in the origin is the same as reflection in the x-axis or y-axis alone, which is not true. Another error is to try to guess the answer by visual intuition instead of applying the coordinate rules. Systematically applying the transformation formulas avoids these issues.

Final Answer:
The original point R has coordinates (5, 1).