Find the transformed equation of
`x^(2)+2sqrt3 xy-y^(2) = 2a^(2)` when the axes are rotated through an angle `30^(0).`

Answer

To find the transformed equation of the given equation \( x^2 + 2\sqrt{3}xy - y^2 = 2a^2 \) when the axes are rotated through an angle of \( 30^\circ \), we will follow these steps:

### Step 1: Define the transformation
We start by defining the transformation equations for the rotation of axes:
\[ x = X \cos \theta - Y \sin \theta \]
\[ y = X \sin \theta + Y \cos \theta \]
where \( \theta = 30^\circ \).

### Step 2: Substitute the values of \( \cos 30^\circ \) and \( \sin 30^\circ \)
Using the trigonometric values:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \]
we can substitute these into the transformation equations:
\[ x = X \cdot \frac{\sqrt{3}}{2} - Y \cdot \frac{1}{2} \]
\[ y = X \cdot \frac{1}{2} + Y \cdot \frac{\sqrt{3}}{2} \]

### Step 3: Substitute \( x \) and \( y \) into the original equation
Now, we substitute \( x \) and \( y \) into the original equation:
\[ \left( X \cdot \frac{\sqrt{3}}{2} - Y \cdot \frac{1}{2} \right)^2 + 2\sqrt{3} \left( X \cdot \frac{\sqrt{3}}{2} - Y \cdot \frac{1}{2} \right)\left( X \cdot \frac{1}{2} + Y \cdot \frac{\sqrt{3}}{2} \right) - \left( X \cdot \frac{1}{2} + Y \cdot \frac{\sqrt{3}}{2} \right)^2 = 2a^2 \]

### Step 4: Expand the equation
Now we will expand each term:
1. **First term**:
\[ \left( X \cdot \frac{\sqrt{3}}{2} - Y \cdot \frac{1}{2} \right)^2 = \frac{3}{4}X^2 - \frac{\sqrt{3}}{2}XY + \frac{1}{4}Y^2 \]
2. **Second term**:
\[ 2\sqrt{3} \left( X \cdot \frac{\sqrt{3}}{2} - Y \cdot \frac{1}{2} \right)\left( X \cdot \frac{1}{2} + Y \cdot \frac{\sqrt{3}}{2} \right) = 2\sqrt{3} \left( \frac{3}{4}XY - \frac{1}{4}Y^2 - \frac{1}{4}X^2 + \frac{\sqrt{3}}{4}XY \right) \]
This simplifies to:
\[ \frac{3\sqrt{3}}{2}XY - \frac{\sqrt{3}}{2}Y^2 - \frac{\sqrt{3}}{2}X^2 \]
3. **Third term**:
\[ \left( X \cdot \frac{1}{2} + Y \cdot \frac{\sqrt{3}}{2} \right)^2 = \frac{1}{4}X^2 + \frac{\sqrt{3}}{2}XY + \frac{3}{4}Y^2 \]

### Step 5: Combine all terms
Combining all the expanded terms:
\[ \frac{3}{4}X^2 - \frac{\sqrt{3}}{2}XY + \frac{1}{4}Y^2 + \frac{3\sqrt{3}}{2}XY - \frac{\sqrt{3}}{2}Y^2 - \left( \frac{1}{4}X^2 + \frac{\sqrt{3}}{2}XY + \frac{3}{4}Y^2 \right) = 2a^2 \]
This results in:
\[ \left( \frac{3}{4} - \frac{1}{4} \right)X^2 + \left( \frac{1}{4} - \frac{3}{4} \right)Y^2 + \left( -\frac{\sqrt{3}}{2} + \frac{3\sqrt{3}}{2} - \frac{\sqrt{3}}{2} \right)XY = 2a^2 \]
This simplifies to:
\[ X^2 - Y^2 = 2a^2 \]

### Final Step: Write the transformed equation
Thus, the transformed equation after the rotation is:
\[ X^2 - Y^2 = a^2 \]