Question

The equation of a line which makes an angle of \( 45^\circ \) with each of the pair of lines \[ xy - x - y + 1 = 0 \] is:

• A. \( x - y = 5 \)
• B. \( 2x + y = 3 \)
• C. \( x + 7y = 8 \)
• D. \( 3x - y = 2 \)

Hint:

For problems involving a line making equal angles with given lines, use the angular bisector method or properties of homogeneous equations to find the required line equation.

Answer 1

The Correct Option is A

Step 1: Understanding the given equation
The given equation of the pair of lines is: \[ xy - x - y + 1 = 0. \] Rewriting it, we can express it as two linear factors, say: \[ (x - a)(y - b) = 0. \] Step 2: Finding the required line
A line making an angle \( 45^\circ \) with both of these lines satisfies the angular bisector equation condition. Using this property, we find: \[ x - y = 5. \] Step 3: Conclusion
Thus, the final answer is: \[ \boxed{x - y = 5}. \]