In coordinate geometry, a point Q(a, b) is first reflected in the y-axis to a point Q1, and then Q1 is reflected in the x-axis to reach the point (6, 2). What are the coordinates of the original point Q?

Introduction / Context:
This question tests the concept of reflections of points in the Cartesian coordinate plane. Specifically, it involves successive reflections of a point in the y-axis and then in the x-axis, and asks you to work backwards to find the original coordinates of the point before any reflection took place.

Given Data / Assumptions:

• Original point is Q(a, b).
• Q is reflected in the y-axis to obtain Q1.
• Q1 is then reflected in the x-axis to obtain the final point (6, 2).
• Standard rules of reflection in coordinate geometry apply.

Concept / Approach:
Reflection in the y-axis changes the sign of the x-coordinate but keeps the y-coordinate the same. Reflection in the x-axis changes the sign of the y-coordinate but keeps the x-coordinate the same. By applying these transformation rules step by step and then equating the final coordinates, we can solve for the original values of a and b.

Step-by-Step Solution:
Step 1: Original point Q has coordinates (a, b). Step 2: Reflect Q in the y-axis. This gives Q1 = (−a, b). Step 3: Reflect Q1 in the x-axis. Reflection in the x-axis changes y to −y, so the new point is (−a, −b). Step 4: We are told that this final point equals (6, 2). Therefore, (−a, −b) = (6, 2). Step 5: Equate coordinates: −a = 6 and −b = 2. Step 6: Solve for a and b: a = −6 and b = −2. Step 7: Hence, the original point Q has coordinates (−6, −2).

Verification / Alternative check:
Start with Q = (−6, −2). Reflect in the y-axis: x changes sign to get Q1 = (6, −2). Then reflect Q1 in the x-axis: y changes sign to get (6, 2), which matches the given final point. This confirms that the original coordinates are correct.

Why Other Options Are Wrong:
Option (−6, 2) would reflect to (6, 2) after only one reflection, not two successive reflections as described.
Option (6, −2) is actually Q1, the intermediate point after the y-axis reflection, not the original point Q.
Option (2, −6) does not produce (6, 2) under the given sequence of reflections.
Option (6, 2) is the final point after both reflections, not the starting point.

Common Pitfalls:
A common mistake is to reverse the order of reflections or to change both coordinates at once without following the correct sequence. Another error is to confuse which axis affects which coordinate. Remember that the y-axis reflection affects only the x-coordinate, and the x-axis reflection affects only the y-coordinate.

Final Answer:
The original point Q has coordinates (−6, −2).