Find the transformed equation of the curve :
`x^(2)+y^(2)+4x-6y+16=0` when the origin is shifted to the point `(-2,3)`.

Answer

To find the transformed equation of the curve given by \( x^2 + y^2 + 4x - 6y + 16 = 0 \) when the origin is shifted to the point \((-2, 3)\), we follow these steps:

### Step 1: Identify the transformation
When we shift the origin from \((0, 0)\) to \((-2, 3)\), we need to express the new coordinates in terms of the old coordinates. The transformation can be expressed as:
\[ x' = x + 2 \quad \text{and} \quad y' = y - 3 \]
where \((x', y')\) are the new coordinates after the shift.

### Step 2: Substitute the new coordinates into the original equation
We substitute \(x = x' - 2\) and \(y = y' + 3\) into the original equation:
\[ (x' - 2)^2 + (y' + 3)^2 + 4(x' - 2) - 6(y' + 3) + 16 = 0 \]

### Step 3: Expand the equation
Now, we will expand each term:
1. Expanding \((x' - 2)^2\):
\[ (x' - 2)^2 = x'^2 - 4x' + 4 \]
2. Expanding \((y' + 3)^2\):
\[ (y' + 3)^2 = y'^2 + 6y' + 9 \]
3. Expanding \(4(x' - 2)\):
\[ 4(x' - 2) = 4x' - 8 \]
4. Expanding \(-6(y' + 3)\):
\[ -6(y' + 3) = -6y' - 18 \]

### Step 4: Combine all the terms
Now we can combine all the expanded terms:
\[ x'^2 - 4x' + 4 + y'^2 + 6y' + 9 + 4x' - 8 - 6y' - 18 + 16 = 0 \]

### Step 5: Simplify the equation
Combining like terms:
- The \(x'\) terms: \(-4x' + 4x' = 0\)
- The \(y'\) terms: \(6y' - 6y' = 0\)
- The constant terms: \(4 + 9 - 8 - 18 + 16 = 3\)

Thus, we have:
\[ x'^2 + y'^2 + 3 = 0 \]

### Step 6: Write the final transformed equation
The transformed equation of the curve is:
\[ x'^2 + y'^2 + 3 = 0 \]