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

Introduction / Context:
This coordinate geometry question involves reflections of a point in the origin and in the Y-axis. You are told that a point P(a, b) is reflected in the origin to give P1, and then P1 is reflected in the Y-axis to produce the final point with coordinates (4, -3). You must determine the original coordinates (a, b). This tests your understanding of transformation rules for reflections in the coordinate plane.

Given Data / Assumptions:

- Original point is P(a, b). - Reflection of P in the origin gives P1. - Reflection of P1 in the Y-axis gives the final point with coordinates (4, -3). - We must find the coordinates (a, b).

Concept / Approach:
Reflections in the coordinate plane follow simple algebraic rules. Reflecting a point (x, y) in the origin changes it to (-x, -y). Reflecting a point (x, y) in the Y-axis changes it to (-x, y). Starting from P(a, b), we apply these transformations step by step: first reflect in the origin, then reflect in the Y-axis. This will express the final coordinates in terms of a and b. Setting these expressions equal to the given final coordinates allows us to solve for a and b directly.

Step-by-Step Solution:
Step 1: Start with the original point P(a, b). Step 2: Reflect P in the origin. The rule for reflection in the origin is (x, y) becomes (-x, -y). Step 3: Therefore, P1, the reflection of P in the origin, has coordinates (-a, -b). Step 4: Next, reflect P1 in the Y-axis. The rule for reflection in the Y-axis is (x, y) becomes (-x, y). Step 5: Apply this rule to P1 = (-a, -b). Its reflection in the Y-axis is (a, -b). Call this final point P2. Step 6: We are told that this final point P2 has coordinates (4, -3). Step 7: Therefore, (a, -b) = (4, -3). Step 8: Equate coordinates: a = 4 and -b = -3. Step 9: From -b = -3, we get b = 3. Step 10: Thus, the original point P has coordinates (4, 3).

Verification / Alternative check:
Verify by performing the transformations explicitly. Start from P(4, 3). Reflect in the origin to get P1(-4, -3). Then reflect P1(-4, -3) in the Y-axis: (x, y) becomes (-x, y), so (-4, -3) becomes (4, -3). This matches the given final point. The transformation chain is therefore correct, confirming that P is indeed (4, 3).

Why Other Options Are Wrong:
Option B (-4, 3) would reflect in the origin to (4, -3) and then in the Y-axis to (-4, -3), not (4, -3). Options C (3, 4) and D (-3, 4) produce entirely different final coordinates after the two reflections. Only (4, 3) results in (4, -3) after reflecting first in the origin and then in the Y-axis.

Common Pitfalls:
Some learners confuse reflection in the origin with reflection in one of the axes, or they apply the transformations in the wrong order. Another common mistake is to think that reflecting in the origin followed by the Y-axis is equivalent to reflecting in the X-axis, which it is not. The safest approach is to apply each transformation one at a time using the correct coordinate rules and then equate the result to the given point.

Final Answer:
The original coordinates of point P are (4, 3), which corresponds to option A.