If `3X^(2)+XY-Y^(2)-7X+Y+7 = 0` is the transformed equation of a curve, 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 \(3X^2 + XY - Y^2 - 7X + Y + 7 = 0\) after translating the axes to the point (1, 2), we will follow these steps:

### Step 1: Identify the new origin
The new origin after translation is given as (h, k) = (1, 2).

### Step 2: Substitute the new coordinates
We need to express the new coordinates \(X\) and \(Y\) in terms of the old coordinates \(x\) and \(y\):
\[ X = x - h = x - 1 \]
\[ Y = y - k = y - 2 \]

### Step 3: Substitute \(X\) and \(Y\) into the transformed equation
Now we substitute \(X\) and \(Y\) into the transformed equation:
\[ 3(x - 1)^2 + (x - 1)(y - 2) - (y - 2)^2 - 7(x - 1) + (y - 2) + 7 = 0 \]

### Step 4: Expand the equation
Now we will expand each term:
1. \(3(x - 1)^2 = 3(x^2 - 2x + 1) = 3x^2 - 6x + 3\)
2. \((x - 1)(y - 2) = xy - 2x - y + 2\)
3. \(-(y - 2)^2 = - (y^2 - 4y + 4) = -y^2 + 4y - 4\)
4. \(-7(x - 1) = -7x + 7\)
5. \((y - 2) = y - 2\)
6. The constant \(+7\) remains as is.

Now, substituting these expansions back into the equation:
\[ 3x^2 - 6x + 3 + xy - 2x - y + 2 - y^2 + 4y - 4 - 7x + 7 + y - 2 + 7 = 0 \]

### Step 5: Combine like terms
Now we will combine all the like terms:
- \(3x^2\)
- The \(y^2\) term: \(-y^2\)
- The \(x\) terms: \(-6x - 2x - 7x = -15x\)
- The \(y\) terms: \(-y + 4y + y = 4y\)
- The constant terms: \(3 + 2 - 4 + 7 - 2 + 7 = 13\)

Putting it all together, we get:
\[ 3x^2 - y^2 - 15x + 4y + xy + 13 = 0 \]

### Step 6: Write the final equation
Thus, the original equation of the curve is:
\[ 3x^2 - y^2 - 15x + 4y + xy + 13 = 0 \]

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