If the origin is shifted to (1,-2) the co -ordinates of A become (2,3). What are the original co ordinates of A?

Answer

To find the original coordinates of point A before the origin was shifted, we can follow these steps:

### Step 1: Understand the Shift of Origin
When the origin is shifted from (0, 0) to (1, -2), the coordinates of any point change accordingly. If the original coordinates of point A are (x, y), then the new coordinates (x', y') with respect to the new origin can be expressed as:
- \( x' = x - \alpha \)
- \( y' = y - \beta \)

Where \( \alpha = 1 \) and \( \beta = -2 \).

### Step 2: Set Up the Equations
Given that the new coordinates of point A are (2, 3) after the shift, we can write:
- \( x' = 2 \)
- \( y' = 3 \)

Substituting the values of \( \alpha \) and \( \beta \) into the equations gives us:
1. \( 2 = x - 1 \)
2. \( 3 = y + 2 \)

### Step 3: Solve for x
From the first equation:
\[ x - 1 = 2 \]
Adding 1 to both sides:
\[ x = 2 + 1 \]
Thus,
\[ x = 3 \]

### Step 4: Solve for y
From the second equation:
\[ y + 2 = 3 \]
Subtracting 2 from both sides:
\[ y = 3 - 2 \]
Thus,
\[ y = 1 \]

### Step 5: Conclusion
The original coordinates of point A are:
\[ (x, y) = (3, 1) \]

### Final Answer
The original coordinates of point A are (3, 1).

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