Question

The combined equation of the lines passing through the point \( (3, 4) \) and each making an angle of 45° with the line \( x + y + 1 = 0 \) is:

• A. \( xy - 4x - 3y + 12 = 0 \)
• B. \( (3x - 2y - 1)(x - 2y + 2) = 0 \)
• C. \( (3x + 2y - 17)(x + 2y - 11) = 0 \)
• D. \( xy - 4x + 3y + 12 = 0 \)

Hint:

When given a common point and angle between two lines, use the combined equation formula to find the equation of both lines.

Answer 1

The Correct Option is A

We are given that the lines pass through the point \( (3, 4) \) and each makes an angle of 45° with the line \( x + y + 1 = 0 \). The combined equation of two lines passing through a common point is given by: \[ (x - x_1)(x_1 + y_1 - 1) = 1 \] Here, the line makes an angle of 45° with the line \( x + y + 1 = 0 \). Therefore, we can use the formula for the combined equation: \[ (x - 3)(y - 4) = 45^\circ \] This equation simplifies to: \[ xy - 4x - 3y + 12 = 0 \] Thus, the correct answer is \( xy - 4x - 3y + 12 = 0 \).