A variable line L passing through the point (2,0) intersects the circle x^2 + y^2 = 16 at points A and B . Let C be the circle drawn with AB as its diameter. If Q is the center of the image of circle C with respect to the line x - y + 1 = 0 , then the locus of Q is :

Step-by-step solution

Let the center of circle C be P(h, k) . Since AB is the diameter of C , P(h, k) is the midpoint of the chord AB of the circle x^2 + y^2 = 16 . The equation of the chord AB with midpoint (h, k) is given by T = S₁ : hx + ky - 16 = h^2 + k^2 - 16 hx + ky = h^2 + k^2 Since the chord AB lies on the variable line L which passes through (2, 0) , we substitute x = 2 and y = 0 into the chord equation: 2h + 0 = h^2 + k^2 h^2 + k^2 - 2h = 0 This is the locus of P . Now, Q( , ) is the reflection of P(h, k) in the line x - y +