If the transformed equation of curve is
`X^(2)+3XY-2Y^(2)+17X-7Y-11=0` when the axes are translated to the point (2,3) then find the original equation of the curve.

Answer

To find the original equation of the curve from the transformed equation given the translation of axes, we can follow these steps:

### Step 1: Identify the transformation
The new coordinates \( X \) and \( Y \) are related to the old coordinates \( x \) and \( y \) by the equations:
\[ X = x - h \quad \text{and} \quad Y = y - k \]
where \( (h, k) \) is the new origin. In this case, \( h = 2 \) and \( k = 3 \).

### Step 2: Substitute the transformations into the equation
We substitute \( X \) and \( Y \) into the transformed equation:
\[ X^2 + 3XY - 2Y^2 + 17X - 7Y - 11 = 0 \]
Substituting \( X = x - 2 \) and \( Y = y - 3 \):
\[ (x - 2)^2 + 3(x - 2)(y - 3) - 2(y - 3)^2 + 17(x - 2) - 7(y - 3) - 11 = 0 \]

### Step 3: Expand each term
Now we will expand each term:
1. \( (x - 2)^2 = x^2 - 4x + 4 \)
2. \( 3(x - 2)(y - 3) = 3(xy - 3x - 2y + 6) = 3xy - 9x - 6y + 18 \)
3. \( -2(y - 3)^2 = -2(y^2 - 6y + 9) = -2y^2 + 12y - 18 \)
4. \( 17(x - 2) = 17x - 34 \)
5. \( -7(y - 3) = -7y + 21 \)

Combining these, we have:
\[ x^2 - 4x + 4 + 3xy - 9x - 6y + 18 - 2y^2 + 12y - 18 + 17x - 34 - 7y + 21 - 11 = 0 \]

### Step 4: Combine like terms
Now, we will combine all the like terms:
- For \( x^2 \): \( x^2 \)
- For \( xy \): \( 3xy \)
- For \( x \): \( -4x - 9x + 17x = 4x \)
- For \( y^2 \): \( -2y^2 \)
- For \( y \): \( -6y + 12y - 7y = -y \)
- Constant terms: \( 4 + 18 - 18 - 34 + 21 - 11 = -20 \)

### Step 5: Write the original equation
Putting it all together, we get the original equation:
\[ x^2 - 2y^2 + 3xy + 4x - y - 20 = 0 \]

### Final Answer
The original equation of the curve is:
\[ x^2 - 2y^2 + 3xy + 4x - y - 20 = 0 \]