Find the equations of the bisectors of the angles formed by the following pairs of lines
`x+sqrt(3)y=6+2sqrt(3)` and `x-sqrt(3)y=6-2sqrt(3)`

Answer

To find the equations of the bisectors of the angles formed by the given lines, we start with the equations of the lines:

1. \( x + \sqrt{3}y = 6 + 2\sqrt{3} \)
2. \( x - \sqrt{3}y = 6 - 2\sqrt{3} \)

We can rewrite these equations in the standard form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \):

1. \( x + \sqrt{3}y - (6 + 2\sqrt{3}) = 0 \)
- Here, \( a_1 = 1 \), \( b_1 = \sqrt{3} \), and \( c_1 = -(6 + 2\sqrt{3}) \).

2. \( x - \sqrt{3}y - (6 - 2\sqrt{3}) = 0 \)
- Here, \( a_2 = 1 \), \( b_2 = -\sqrt{3} \), and \( c_2 = -(6 - 2\sqrt{3}) \).

Now, we can use the formula for the angle bisectors of two lines given by:

\[ \frac{a_1x + b_1y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2x + b_2y + c_2}{\sqrt{a_2^2 + b_2^2}} \]

Calculating \( \sqrt{a_1^2 + b_1^2} \) and \( \sqrt{a_2^2 + b_2^2} \):

\[ \sqrt{a_1^2 + b_1^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

\[ \sqrt{a_2^2 + b_2^2} = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

Now substituting these values into the angle bisector formula:

\[ \frac{x + \sqrt{3}y - (6 + 2\sqrt{3})}{2} = \pm \frac{x - \sqrt{3}y - (6 - 2\sqrt{3})}{2} \]

We can simplify this to:

\[ x + \sqrt{3}y - (6 + 2\sqrt{3}) = \pm (x - \sqrt{3}y - (6 - 2\sqrt{3})) \]

This gives us two cases to solve:

### Case 1: Positive Sign

\[ x + \sqrt{3}y - (6 + 2\sqrt{3}) = x - \sqrt{3}y - (6 - 2\sqrt{3}) \]

Cancelling \( x \) from both sides:

\[ \sqrt{3}y - (6 + 2\sqrt{3}) = -\sqrt{3}y - (6 - 2\sqrt{3}) \]

Rearranging gives:

\[ \sqrt{3}y + \sqrt{3}y = 6 - 2\sqrt{3} + 6 + 2\sqrt{3} \]

\[ 2\sqrt{3}y = 12 \]

\[ y = 6/\sqrt{3} = 2 \]

### Case 2: Negative Sign

\[ x + \sqrt{3}y - (6 + 2\sqrt{3}) = - (x - \sqrt{3}y - (6 - 2\sqrt{3})) \]

This simplifies to:

\[ x + \sqrt{3}y - (6 + 2\sqrt{3}) = -x + \sqrt{3}y + (6 - 2\sqrt{3}) \]

Cancelling \( \sqrt{3}y \) from both sides:

\[ x - (6 + 2\sqrt{3}) = -x + (6 - 2\sqrt{3}) \]

Rearranging gives:

\[ x + x = 6 - 2\sqrt{3} + 6 + 2\sqrt{3} \]

\[ 2x = 12 \]

\[ x = 6 \]

### Final Result

The equations of the angle bisectors are:

1. \( y = 2 \)
2. \( x = 6 \)