Find the new co-ordinates of the following points when origin is shifted to the point `(-1,4)` :
`(i) (2,5)`
`(ii) (-3,-2)`
`(iii) (1,-4)`

Answer

To find the new coordinates of the given points when the origin is shifted to the point (-1, 4), we will use the following transformation rules:

1. Let the old coordinates be represented as (x, y).
2. Let the new coordinates be represented as (X, Y).
3. The transformation when the origin is shifted to the point (h, k) is given by:
- \( X = x + h \)
- \( Y = y + k \)

In our case, the new origin is at (-1, 4), so h = -1 and k = 4.

Now, we will calculate the new coordinates for each of the given points.

### (i) For the point (2, 5):
- Old coordinates: \( x = 2 \), \( y = 5 \)
- Applying the transformation:
- \( X = x + h = 2 + (-1) = 2 - 1 = 1 \)
- \( Y = y + k = 5 + 4 = 5 + 4 = 9 \)
- New coordinates: \( (X, Y) = (1, 9) \)

### (ii) For the point (-3, -2):
- Old coordinates: \( x = -3 \), \( y = -2 \)
- Applying the transformation:
- \( X = x + h = -3 + (-1) = -3 - 1 = -4 \)
- \( Y = y + k = -2 + 4 = -2 + 4 = 2 \)
- New coordinates: \( (X, Y) = (-4, 2) \)

### (iii) For the point (1, -4):
- Old coordinates: \( x = 1 \), \( y = -4 \)
- Applying the transformation:
- \( X = x + h = 1 + (-1) = 1 - 1 = 0 \)
- \( Y = y + k = -4 + 4 = -4 + 4 = 0 \)
- New coordinates: \( (X, Y) = (0, 0) \)

### Summary of New Coordinates:
1. For the point (2, 5), the new coordinates are (1, 9).
2. For the point (-3, -2), the new coordinates are (-4, 2).
3. For the point (1, -4), the new coordinates are (0, 0).

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