If the transformed equation of curve is
`X^(2)+2Y^(2)+16=0` 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 \(X^{2} + 2Y^{2} + 16 = 0\) and the translation of axes to the point \((-1, 2)\), we can follow these steps:

### Step 1: Understand the Translation of Axes
When the axes are translated to a new origin \((H, K)\), the relationships between the new coordinates \((X, Y)\) and the old coordinates \((x, y)\) are given by:
\[ X = x - H \quad \text{and} \quad Y = y - K \]
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, we substitute \(X\) and \(Y\) into the transformed equation:
\[ X^{2} + 2Y^{2} + 16 = 0 \]
Substituting for \(X\) and \(Y\):
\[ (x + 1)^{2} + 2(y - 2)^{2} + 16 = 0 \]

### Step 4: Expand the Equation
Now, we expand the equation:
1. Expand \((x + 1)^{2}\):
\[ (x + 1)^{2} = x^{2} + 2x + 1 \]
2. Expand \(2(y - 2)^{2}\):
\[ 2(y - 2)^{2} = 2(y^{2} - 4y + 4) = 2y^{2} - 8y + 8 \]
3. Combine all parts:
\[ x^{2} + 2x + 1 + 2y^{2} - 8y + 8 + 16 = 0 \]

### Step 5: Simplify the Equation
Combine like terms:
\[ x^{2} + 2y^{2} + 2x - 8y + (1 + 8 + 16) = 0 \]
This simplifies to:
\[ x^{2} + 2y^{2} + 2x - 8y + 25 = 0 \]

### Final Step: Write the Original Equation
Thus, the original equation of the curve is:
\[ x^{2} + 2y^{2} + 2x - 8y + 25 = 0 \]

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