Consider the circle C: x^2+y^2-6x-8y-11=0 . Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle x^2+y^2- x - y - = 0 , then + + 2 is equal to ________.

Step-by-step solution

Let the foot of the perpendicular from the origin to the chord AB be P(h, k) . The slope of OP is k h . Since the chord AB is perpendicular to OP , its equation is given by: y - k = - h k (x - h) hx + ky = h^2 + k^2 This can be rewritten as hx + ky h^2 + k^2 = 1 . The chord AB subtends a right angle at the origin. We homogenize the equation of the circle x^2 + y^2 - 6x - 8y - 11 = 0 with the equation of the chord to find the joint equation of the lines OA and OB : x^2 + y^2 - (6x + 8y) ( hx + ky h^2 + k^2 ) - 11 (