If the transformed equation of a curve is `17x^(2) - 16xy + 17y^(2)=225` when the axes are rotated through an angle `45^(@)`, then the original equation of the curve is

Answer

To find the original equation of the curve given the transformed equation after rotating the axes by \(45^\circ\), we can follow these steps:

### Step 1: Understand the transformation equations
When the axes are rotated through an angle \(\theta\), the new coordinates \(x'\) and \(y'\) are related to the original coordinates \(x\) and \(y\) by the following equations:
\[ x' = x \cos \theta + y \sin \theta \]
\[ y' = -x \sin \theta + y \cos \theta \]

For \(\theta = 45^\circ\), we have:
\[ \cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}} \]
Thus, the equations become:
\[ x' = \frac{x + y}{\sqrt{2}} \]
\[ y' = \frac{-x + y}{\sqrt{2}} \]

### Step 2: Substitute \(x'\) and \(y'\) into the transformed equation
The transformed equation given is:
\[ 17x^2 - 16xy + 17y^2 = 225 \]
We will substitute \(x'\) and \(y'\) into this equation.

### Step 3: Substitute the expressions for \(x'\) and \(y'\)
Substituting \(x' = \frac{x + y}{\sqrt{2}}\) and \(y' = \frac{-x + y}{\sqrt{2}}\) into the transformed equation:
\[ 17\left(\frac{x + y}{\sqrt{2}}\right)^2 - 16\left(\frac{x + y}{\sqrt{2}}\right)\left(\frac{-x + y}{\sqrt{2}}\right) + 17\left(\frac{-x + y}{\sqrt{2}}\right)^2 = 225 \]

### Step 4: Simplify the equation
Calculating each term:
1. For the first term:
\[ 17\left(\frac{x + y}{\sqrt{2}}\right)^2 = 17 \cdot \frac{x^2 + 2xy + y^2}{2} = \frac{17}{2}(x^2 + 2xy + y^2) \]

2. For the second term:
\[ -16\left(\frac{x + y}{\sqrt{2}}\right)\left(\frac{-x + y}{\sqrt{2}}\right) = -16 \cdot \frac{(x + y)(-x + y)}{2} = -8(-x^2 + y^2 + xy - xy) = 8(y^2 - x^2) \]

3. For the third term:
\[ 17\left(\frac{-x + y}{\sqrt{2}}\right)^2 = 17 \cdot \frac{x^2 - 2xy + y^2}{2} = \frac{17}{2}(x^2 - 2xy + y^2) \]

### Step 5: Combine all terms
Combining all the terms gives:
\[ \frac{17}{2}(x^2 + 2xy + y^2) + 8(y^2 - x^2) + \frac{17}{2}(x^2 - 2xy + y^2) = 225 \]

### Step 6: Collect like terms
Combining like terms:
- The \(x^2\) terms: \(\frac{17}{2}x^2 - 8x^2 + \frac{17}{2}x^2 = \frac{34}{2}x^2 - 8x^2 = 17x^2 - 8x^2 = 9x^2\)
- The \(y^2\) terms: \(\frac{17}{2}y^2 + 8y^2 + \frac{17}{2}y^2 = \frac{34}{2}y^2 + 8y^2 = 17y^2 + 8y^2 = 25y^2\)
- The \(xy\) terms cancel out.

Thus, we have:
\[ 9x^2 + 25y^2 = 450 \]

### Step 7: Divide by 2
To simplify, divide the entire equation by 2:
\[ 25x^2 + 9y^2 = 225 \]

### Final Answer
The original equation of the curve is:
\[ 25x^2 + 9y^2 = 225 \]