If the transformed equation of curve is
`X^(2)+Y^(2)=4` when the axes are translated to the point (-1,2) then find the original equation of the curve.

Answer

To find the original equation of the curve given the transformed equation \( X^2 + Y^2 = 4 \) after translating the axes to the point (-1, 2), we can follow these steps:

### Step 1: Understand the Translation of Axes
When the axes are translated, the new coordinates \( (X, Y) \) are related to the old coordinates \( (x, y) \) by the equations:
\[ X = x + h \quad \text{and} \quad Y = y + k \]
where \( (h, k) \) is the point to which the axes are translated. In this case, \( h = -1 \) and \( k = 2 \).

### Step 2: Substitute the Translated Coordinates
Substituting \( h \) and \( k \) into the equations gives:
\[ X = x + 1 \quad \text{and} \quad Y = y - 2 \]

### Step 3: Substitute into the Transformed Equation
Now substitute \( X \) and \( Y \) into the transformed equation \( X^2 + Y^2 = 4 \):
\[ (x + 1)^2 + (y - 2)^2 = 4 \]

### Step 4: Expand the Equation
Now, expand both squares:
\[ (x + 1)^2 = x^2 + 2x + 1 \]
\[ (y - 2)^2 = y^2 - 4y + 4 \]
So, substituting these into the equation gives:
\[ x^2 + 2x + 1 + y^2 - 4y + 4 = 4 \]

### Step 5: Simplify the Equation
Combine like terms:
\[ x^2 + y^2 + 2x - 4y + 5 = 4 \]
Now, subtract 4 from both sides:
\[ x^2 + y^2 + 2x - 4y + 1 = 0 \]

### Step 6: Final Result
Thus, the original equation of the curve is:
\[ x^2 + y^2 + 2x - 4y + 1 = 0 \]