Transform the equation ` 2 x ^(2) + 3 sqrt"" 3 xy + 3y ^(2) = 2 ` from axes inclined at `30^(@)` to rectangular axes of x remaining unchanged

Answer

To transform the equation \( 2x^2 + 3\sqrt{3}xy + 3y^2 = 2 \) from axes inclined at \( 30^\circ \) to rectangular axes where \( x \) remains unchanged, we will follow these steps:

### Step 1: Identify the transformation equations
We need to use the transformation equations for coordinates when changing from one set of axes to another. The transformation from the inclined axes to the rectangular axes can be expressed as:
\[ x = x' \cos(30^\circ) + y' \sin(30^\circ) \]
\[ y = -x' \sin(30^\circ) + y' \cos(30^\circ) \]
where \( \theta = 30^\circ \).

### Step 2: Substitute the values of \( \cos(30^\circ) \) and \( \sin(30^\circ) \)
Using the known values:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \]
we can rewrite 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 these expressions for \( x \) and \( y \) into the original equation \( 2x^2 + 3\sqrt{3}xy + 3y^2 = 2 \).

First, calculate \( x^2 \):
\[ x^2 = \left( x' \cdot \frac{\sqrt{3}}{2} + y' \cdot \frac{1}{2} \right)^2 = \frac{3}{4}x'^2 + \frac{\sqrt{3}}{2}x'y' + \frac{1}{4}y'^2 \]

Next, calculate \( y^2 \):
\[ y^2 = \left( -x' \cdot \frac{1}{2} + y' \cdot \frac{\sqrt{3}}{2} \right)^2 = \frac{1}{4}x'^2 - \frac{\sqrt{3}}{2}x'y' + \frac{3}{4}y'^2 \]

Now calculate \( xy \):
\[ xy = \left( x' \cdot \frac{\sqrt{3}}{2} + y' \cdot \frac{1}{2} \right) \left( -x' \cdot \frac{1}{2} + y' \cdot \frac{\sqrt{3}}{2} \right) \]
Expanding this gives:
\[ xy = -\frac{\sqrt{3}}{4}x'^2 + \frac{3}{4}y'^2 + \frac{1}{4}x'y' \]

### Step 4: Substitute into the original equation
Now substitute \( x^2 \), \( y^2 \), and \( xy \) into the original equation:
\[ 2\left(\frac{3}{4}x'^2 + \frac{\sqrt{3}}{2}x'y' + \frac{1}{4}y'^2\right) + 3\sqrt{3}\left(-\frac{\sqrt{3}}{4}x'^2 + \frac{3}{4}y'^2 + \frac{1}{4}x'y'\right) + 3\left(\frac{1}{4}x'^2 - \frac{\sqrt{3}}{2}x'y' + \frac{3}{4}y'^2\right) = 2 \]

### Step 5: Simplify the equation
After substituting and simplifying, we will collect like terms:
- Combine \( x'^2 \) terms,
- Combine \( y'^2 \) terms,
- Combine \( x'y' \) terms.

This will yield a new equation in terms of \( x' \) and \( y' \).

### Step 6: Final equation
After simplification, we will arrive at the transformed equation in the form:
\[ Ax'^2 + Bx'y' + Cy'^2 = D \]
where \( A, B, C, D \) are constants derived from the simplification.