Question

If two pairs of lines \( x^2 - 2mxy - y^2 = 0 \) and \( x^2 - 2nxy - y^2 = 0 \) are such that one of them represents the bisector of the angles between the other, then:

• A. \( mn = 1 \)
• B. \( m + n = mn \)
• C. \( mn = -1 \)
• D. \( m - n = mn \)

Hint:

When solving problems involving angle bisectors, use the condition that the product of the slopes of the bisectors equals \( -1 \).

Answer 1

The Correct Option is C

We are given two pairs of lines represented by the equations: \[ x^2 - 2mxy - y^2 = 0 \quad \text{and} \quad x^2 - 2nxy - y^2 = 0 \] These equations represent two pairs of straight lines, and we are told that one pair represents the bisector of the angles between the other pair. Step 1: Use the condition for the angle bisector For the two lines to be angle bisectors, the product of the slopes \( m \) and \( n \) must be equal to \( -1 \). Thus, we have the condition: \[ mn = -1 \] Thus, the correct answer is \( mn = -1 \).