The number of circles passing through the origin and touching the lines x + y = 1 and x - y = 1 is

Step-by-step solution

The given lines are x + y - 1 = 0 and x - y - 1 = 0 . The center of a circle touching both lines must lie on one of their angle bisectors. The equations of the angle bisectors are given by: x + y - 1 1^2 + 1^2 = x - y - 1 1^2 + (-1)^2 x + y - 1 2 = x - y - 1 2 Taking the positive sign, we get 2y = 0 y = 0 . Taking the negative sign, we get 2x - 2 = 0 x = 1 . Case 1: Let the center lie on the bisector y = 0 . Let the center be (h, 0) . Since the circle passes through the origin (0, 0) , its radius is r = (h-0)^2 + (