Find the transformed equation of
`3x^(2)+10xy +3y^(2) = 9` when the axes are rotated through an angle `pi/4`

Answer

To find the transformed equation of the given equation \(3x^2 + 10xy + 3y^2 = 9\) when the axes are rotated through an angle \(\frac{\pi}{4}\), we will follow these steps:

### Step 1: Define the transformation equations
When the axes are rotated by an angle \(\theta\), the transformations for the coordinates are given by:
\[ x = x' \cos \theta - y' \sin \theta \]
\[ y = x' \sin \theta + y' \cos \theta \]
For \(\theta = \frac{\pi}{4}\), we have:
\[ \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]
Thus, the transformations become:
\[ x = \frac{1}{\sqrt{2}}(x' - y') \]
\[ y = \frac{1}{\sqrt{2}}(x' + y') \]

### Step 2: Substitute the transformations into the original equation
We substitute \(x\) and \(y\) into the original equation \(3x^2 + 10xy + 3y^2 = 9\):
\[ 3\left(\frac{1}{\sqrt{2}}(x' - y')\right)^2 + 10\left(\frac{1}{\sqrt{2}}(x' - y')\right)\left(\frac{1}{\sqrt{2}}(x' + y')\right) + 3\left(\frac{1}{\sqrt{2}}(x' + y')\right)^2 = 9 \]

### Step 3: Expand each term
Now we will expand each term:
1. For \(3\left(\frac{1}{\sqrt{2}}(x' - y')\right)^2\):
\[ 3\cdot\frac{1}{2}(x' - y')^2 = \frac{3}{2}(x'^2 - 2x'y' + y'^2) = \frac{3}{2}x'^2 - 3x'y' + \frac{3}{2}y'^2 \]

2. For \(10\left(\frac{1}{\sqrt{2}}(x' - y')\right)\left(\frac{1}{\sqrt{2}}(x' + y')\right)\):
\[ 10\cdot\frac{1}{2}(x'^2 - y'^2) = 5(x'^2 - y'^2) = 5x'^2 - 5y'^2 \]

3. For \(3\left(\frac{1}{\sqrt{2}}(x' + y')\right)^2\):
\[ 3\cdot\frac{1}{2}(x' + y')^2 = \frac{3}{2}(x'^2 + 2x'y' + y'^2) = \frac{3}{2}x'^2 + 3x'y' + \frac{3}{2}y'^2 \]

### Step 4: Combine all the terms
Now, we combine all the expanded terms:
\[ \left(\frac{3}{2}x'^2 - 3x'y' + \frac{3}{2}y'^2\right) + \left(5x'^2 - 5y'^2\right) + \left(\frac{3}{2}x'^2 + 3x'y' + \frac{3}{2}y'^2\right) = 9 \]
Combining like terms:
\[ \left(\frac{3}{2} + 5 + \frac{3}{2}\right)x'^2 + \left(-3 + 3\right)x'y' + \left(\frac{3}{2} - 5 + \frac{3}{2}\right)y'^2 = 9 \]
This simplifies to:
\[ 8x'^2 - 2y'^2 = 9 \]

### Step 5: Final transformed equation
Thus, the transformed equation is:
\[ 8x'^2 - 2y'^2 = 9 \]
We can replace \(x'\) and \(y'\) back with \(x\) and \(y\) since they are just variables:
\[ 8x^2 - 2y^2 = 9 \]