If P is a point on the circle x^2 + y^2 = 16 , Q is a point on the straight line x - my + 3 = 0 (where m > 0 ) and x - y + 2 = 0 is the perpendicular bisector of PQ , then the sum of the abscissae of all such points P is -3 . The value of m is _ _ _ _ .

Step-by-step solution

Let P (h, k) . The reflection of P across the line x - y + 2 = 0 is Q . x - h 1 = y - k -1 = -2 h - k + 2 1^2 + (-1)^2 = -h + k - 2 x = k - 2 and y = h + 2 So, Q (k - 2, h + 2) . Since Q lies on the line x - my + 3 = 0 , (k - 2) - m(h + 2) + 3 = 0 k = mh + 2m - 1 Since P(h, k) lies on the circle x^2 + y^2 = 16 , h^2 + (mh + 2m - 1)^2 = 16 (1 + m^2)h^2 + 2m(2m - 1)h + (2m - 1)^2 - 16 = 0 The sum of the abscissae of all such points P is given as -3 . - 4m^2 - 2m 1 + m^2 = -3 4m^2 - 2m = 3 + 3m^2 m^2 - 2m - 3 = 0 (m -