The point P(a, b) is first reflected in the origin to obtain P1, and then P1 is reflected in the Y axis to give the point (4, −3). What are the coordinates of the original point P?

Introduction / Context:
This question tests your understanding of geometric transformations, especially reflections in the coordinate plane. Reflecting a point in the origin and then in the Y axis changes the signs of the coordinates in a predictable way. You must track these steps in reverse to find the original point P.

Given Data / Assumptions:

- Original point is P(a, b).
- P is reflected in the origin to get P1.
- P1 is then reflected in the Y axis to get the point (4, −3).
- We assume standard Cartesian coordinates with X and Y axes perpendicular.

Concept / Approach:
Reflection in the origin sends any point (x, y) to (−x, −y). Reflection in the Y axis sends (x, y) to (−x, y). To find the original point P, we use these transformation rules. We know the final result, so we apply the transformations in reverse order or express everything symbolically and solve for a and b.

Step-by-Step Solution:
Step 1: Start from the original point P with coordinates (a, b).Step 2: Reflect P in the origin to obtain P1. The reflection in the origin changes both signs, so P1 = (−a, −b).Step 3: Reflect P1 in the Y axis. Reflection in the Y axis changes the sign of the x coordinate but keeps the y coordinate unchanged. Thus the final point is (a, −b).Step 4: We are told that this final point equals (4, −3). Therefore, we can equate coordinates: a = 4 and −b = −3.Step 5: From −b = −3 it follows that b = 3. Thus the original point P has coordinates (4, 3).

Verification / Alternative check:
Take P(4, 3). Reflect in the origin: you get P1(−4, −3). Reflect P1 in the Y axis: x changes sign and y stays the same, so (−4, −3) becomes (4, −3). This matches the given final point, which confirms the answer.

Why Other Options Are Wrong:
- (−4, 3) would reflect in the origin to (4, −3) and then in the Y axis to (−4, −3), which does not match (4, −3).
- (−3, 4) and (3, −4) similarly give different final points when the sequence of reflections is applied.
- (4, −3) is actually the final point after transformations, not the original point P.

Common Pitfalls:
Students sometimes mix up the order of reflections or the effect of each reflection. Remember that origin reflection changes both signs, while Y axis reflection changes only the x sign. Also ensure that you are tracing from the original point to the final point correctly or reversing the steps with care.

Final Answer:
The coordinates of the original point P are (4, 3).