If the combined equation of angle bisectors of the lines x^2 - 2pxy - y^2 = 0 is x^2 - 2qxy - y^2 = 0 , then which of the following is true?

Step-by-step solution

The given equation of the pair of lines is x^2 - 2pxy - y^2 = 0 . Comparing this with the standard equation ax^2 + 2hxy + by^2 = 0 , we get a = 1 , h = -p , and b = -1 . The combined equation of the angle bisectors is given by the formula x^2 - y^2 a - b = xy h . Substituting the values of a , h , and b , we get: x^2 - y^2 1 - (-1) = xy -p x^2 - y^2 2 = xy -p -p(x^2 - y^2) = 2xy px^2 + 2xy - py^2 = 0 Dividing the entire equation by p , we obtain: x^2 + 2 p xy - y^2 = 0 It is given that the combined equation of the