Two lines represented by the equation.`x^2-y^2-2x+1=0` are rotated about the point `(1,0)`, the line making the bigger angle with the position direction ofthe x-axis being turned by `45^@` in the clockwise sense and the other line being turned by `15^@` in the anti-clockwise sense. The combined equation of the pair of lines in their new position is

Options

A. `sqrt3x^(2)-xy+2sqrt3x-y+sqrt3=0`
B. `sqrt3x^(2)-xy-2sqrt3x+y+sqrt3=0`
C. `sqrt3x^(2)-xy-2sqrt3x+sqrt3=0`
D. none of these

Explanation

To solve the problem step by step, we will first rewrite the given equation of the pair of lines, find the slopes of the lines, rotate them according to the specified angles, and then derive the combined equation of the new lines.

### Step 1: Rewrite the given equation
The given equation is:
\[ x^2 - y^2 - 2x + 1 = 0 \]
We can rewrite it as:
\[ (x - 1)^2 - y^2 = 0 \]
This can be factored using the difference of squares:
\[ (x - 1 - y)(x - 1 + y) = 0 \]
Thus, we have two lines:
1. \( y = x - 1 \)
2. \( y = -x + 1 \)