When the origin is shifted to (4,-5) by the translation of axes, find the coordinates of the point
(-2,4) with reference to new axes.

Answer

To solve the problem of finding the coordinates of the point (-2, 4) with respect to the new axes after shifting the origin to (4, -5), we can follow these steps:

### Step 1: Understand the Shift in Origin
The original coordinates of the point are given as (-2, 4). We need to translate these coordinates based on the new origin (4, -5).

### Step 2: Define the Shift
The shift in the x-coordinate (Δx) is 4 (the new x-coordinate of the origin) and the shift in the y-coordinate (Δy) is -5 (the new y-coordinate of the origin).

### Step 3: Calculate the New Coordinates
To find the new coordinates (X, Y) in the new system, we use the following formulas:
- \( X = x + \Delta x \)
- \( Y = y + \Delta y \)

Substituting the values:
- Original x-coordinate (x) = -2
- Original y-coordinate (y) = 4
- Shift in x (Δx) = 4
- Shift in y (Δy) = -5

Calculating X:
\[ X = -2 + 4 = 2 \]

Calculating Y:
\[ Y = 4 - 5 = -1 \]

### Step 4: Write the Final Coordinates
The new coordinates of the point (-2, 4) with respect to the new axes are (2, -1).

### Final Answer
The coordinates of the point (-2, 4) with reference to the new axes are (2, -1).

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