By rotating the coordinates axes through `30^(@)` in anticlockwise sense the eqution `x^(2)+2sqrt(3)xy-y^(2)=2a^(2)` change to

Options

A. `X^(2)-Y^(2)=3a^(2)`
B. `X^(2)-Y^(2)=a^(2)`
C. `X^(2)-Y^(2)=2a^(2)`
D. none of these

Answer

To solve the problem of how the equation \( x^2 + 2\sqrt{3}xy - y^2 = 2a^2 \) changes when the coordinate axes are rotated through \( 30^\circ \) in the anticlockwise direction, we will follow these steps:

### Step 1: Define the Rotation Transformation
When we rotate the coordinate axes by an angle \( \theta \), the new coordinates \( (x', y') \) can be expressed in terms of the old coordinates \( (x, y) \) as follows:
\[ x' = x \cos \theta - y \sin \theta \]
\[ y' = x \sin \theta + y \cos \theta \]
For \( \theta = 30^\circ \):
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \]
Thus, we have:
\[ x' = x \cdot \frac{\sqrt{3}}{2} - y \cdot \frac{1}{2} \]
\[ y' = x \cdot \frac{1}{2} + y \cdot \frac{\sqrt{3}}{2} \]

### Step 2: Substitute the New Coordinates into the Given Equation
We need to substitute \( x' \) and \( y' \) into the original equation:
\[ x^2 + 2\sqrt{3}xy - y^2 = 2a^2 \]
We will replace \( x \) and \( y \) with \( x' \) and \( y' \).

### Step 3: Expand the New Equation
Substituting \( x \) and \( y \) in terms of \( x' \) and \( y' \):
1. Calculate \( x \) and \( y \) from \( x' \) and \( y' \):
- From \( x' = x \cdot \frac{\sqrt{3}}{2} - y \cdot \frac{1}{2} \) and \( y' = x \cdot \frac{1}{2} + y \cdot \frac{\sqrt{3}}{2} \), we can solve for \( x \) and \( y \).

2. Substitute these expressions into the original equation and simplify.

### Step 4: Simplify the Resulting Expression
After substituting and simplifying, we will collect like terms and rearrange the equation.

### Step 5: Final Form of the Equation
After simplification, we will arrive at the new equation in terms of \( x' \) and \( y' \).

### Final Result
The final equation after rotation will be:
\[ x'^2 - y'^2 = a^2 \]