If the pair of lines `sqrt(3)x^2-4x y+sqrt(3)y^2=0` is rotated about the origin by `pi/6` in the anticlockwise sense, then find the equation of the pair in the new position.

Explanation

To find the equation of the pair of lines after rotating the given lines about the origin by \(\frac{\pi}{6}\) (or 30 degrees) in the anticlockwise direction, we can follow these steps:

### Step 1: Write the given equation
The given equation of the pair of lines is:
\[ \sqrt{3}x^2 - 4xy + \sqrt{3}y^2 = 0 \]

### Step 2: Factor the equation
We can factor the equation to find the slopes of the lines. The factored form is:
\[ (\sqrt{3}x - y)(x - \sqrt{3}y) = 0 \]
This gives us the equations of the lines:
1. \(y = \sqrt{3}x\) (slope = \(\sqrt{3}\))
2. \(y = \frac{1}{\sqrt{3}}x\) (slope = \(\frac{1}{\sqrt{3}}\))

### Step 3: Identify the angles of the lines
The slopes correspond to angles:
- For \(y = \sqrt{3}x\), the angle is \(60^\circ\) (or \(\frac{\pi}{3}\)).
- For \(y = \frac{1}{\sqrt{3}}x\), the angle is \(30^\circ\) (or \(\frac{\pi}{6}\)).

### Step 4: Rotate the angles
To rotate the lines by \(\frac{\pi}{6}\) (30 degrees), we add \(\frac{\pi}{6}\) to each angle:
1. New angle for \(y = \sqrt{3}x\):
\[ 60^\circ + 30^\circ = 90^\circ \]
(slope becomes undefined, vertical line)

2. New angle for \(y = \frac{1}{\sqrt{3}}x\):
\[ 30^\circ + 30^\circ = 60^\circ \]
(slope = \(\sqrt{3}\))

### Step 5: Write the new equations of the lines
The new equations of the lines after rotation are:
1. The line corresponding to \(90^\circ\) (vertical line):
\[ x = 0 \]

2. The line corresponding to \(60^\circ\):
\[ y = \sqrt{3}x \]

### Step 6: Formulate the new equation
The new equation of the pair of lines can be expressed as:
\[ x \cdot (y - \sqrt{3}x) = 0 \]
This simplifies to:
\[ x = 0 \quad \text{or} \quad y = \sqrt{3}x \]

### Final Equation
Thus, the final equation of the pair of lines after rotation is:
\[ x(y - \sqrt{3}x) = 0 \]