If two pairs of lines x²-2 m x y-y²=0 and x²-2 n x y-y²=0 are such that one of them represents the bisector of the angles between the other, then

Step-by-step solution

The equation of the pair of lines is x² - 2mxy - y² = 0 . The angle bisectors of this pair are given by the formula x² - y² a - b = xy h . Here a = 1 , b = -1 , and h = -m . Substituting these values, we get x² - y² 1 - (-1) = xy -m , which simplifies to x² - y² 2 = xy -m . Rearranging this gives -m(x² - y²) = 2xy , or mx² + 2xy - my² = 0 . Dividing by -m (assuming m 0 ), we get x² + 2 m xy - y² = 0 . We are given that the other pair of lines is x² - 2nxy - y² = 0 . Comparing the two equations, we must have -2n = 2