Find the transformed equation of the curve `y^(2)-4x+4y+8=0` when the origin is shifted to `(1,-2)`.

Answer

To find the transformed equation of the curve \( y^2 - 4x + 4y + 8 = 0 \) when the origin is shifted to \( (1, -2) \), we will follow these steps:

### Step 1: Substitute the new coordinates
When we shift the origin to \( (1, -2) \), we need to express the new coordinates in terms of the old coordinates. The transformations are:
- \( x = x' + 1 \)
- \( y = y' - 2 \)

Here, \( (x', y') \) are the new coordinates after the shift.

### Step 2: Substitute into the original equation
Now, we will substitute \( x \) and \( y \) in the original equation:
\[ y^2 - 4x + 4y + 8 = 0 \]
Substituting \( x = x' + 1 \) and \( y = y' - 2 \):
\[ (y' - 2)^2 - 4(x' + 1) + 4(y' - 2) + 8 = 0 \]

### Step 3: Expand the equation
Now we will expand the equation:
1. Expand \( (y' - 2)^2 \):
\[ (y' - 2)^2 = y'^2 - 4y' + 4 \]
2. Substitute this back into the equation:
\[ y'^2 - 4y' + 4 - 4(x' + 1) + 4(y' - 2) + 8 = 0 \]
3. Simplify:
\[ y'^2 - 4y' + 4 - 4x' - 4 + 4y' - 8 + 8 = 0 \]
Combine like terms:
\[ y'^2 - 4x' = 0 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ y'^2 = 4x' \]

### Final Transformed Equation
Thus, the transformed equation of the curve when the origin is shifted to \( (1, -2) \) is:
\[ y'^2 = 4x' \]