Let a triangle be formed by the pair of straight lines x^2 - y^2 = 0 and the line x = 6 . The distance between the centroid and the orthocenter of this triangle is

Step-by-step solution

The given pair of straight lines is x^2 - y^2 = 0 , which can be factored as (x - y)(x + y) = 0 . The equations of the three sides of the triangle are x - y = 0 , x + y = 0 , and x = 6 . Solving these equations pairwise, the vertices of the triangle are O(0, 0) , A(6, 6) , and B(6, -6) . The lines x - y = 0 and x + y = 0 are perpendicular to each other (the product of their slopes is 1 (-1) = -1 ). Thus, the triangle is right-angled at the origin O(0, 0) . For a right-angled triangle, the orthocenter lies at the ve