When the origin is shifted to (4,-5) by the translation of axes, find the coordinates of the point (0,3) with reference to the new axes.

Answer

To solve the problem of finding the coordinates of the point (0, 3) with respect to the new axes after shifting the origin to (4, -5), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Old and New Origin**:
- The old origin (0, 0) is the reference point before the shift.
- The new origin is given as (4, -5).

2. **Define the Coordinates of the Point**:
- The coordinates of the point we are interested in are (0, 3) with respect to the old origin.

3. **Use the Translation of Axes Formula**:
- The formulas for translating the axes are:
- \( X = x - H \)
- \( Y = y - K \)
- Here, \( H \) and \( K \) are the coordinates of the new origin. In this case, \( H = 4 \) and \( K = -5 \).

4. **Substitute the Values**:
- For the point (0, 3):
- \( x = 0 \)
- \( y = 3 \)
- Substitute these values into the formulas:
- \( X = 0 - 4 = -4 \)
- \( Y = 3 - (-5) = 3 + 5 = 8 \)

5. **Write the New Coordinates**:
- The coordinates of the point (0, 3) with respect to the new origin (4, -5) are:
- \( (X, Y) = (-4, 8) \)

### Final Answer:
The coordinates of the point (0, 3) with respect to the new axes are **(-4, 8)**.