A fixed point P(0, 1) lies on the circle x^2 + y^2 = 1 . The set of all values of the parameter c for which the line x - y + c = 0 bisects two distinct chords drawn from P to the circle, is equal to :

Step-by-step solution

Let the midpoint of the chord be M(t, t+c) since it lies on the given line x - y + c = 0 . Let the other endpoint of the chord be Q . Since M is the midpoint of P(0, 1) and Q , we have Q = 2M - P = (2t, 2t+2c-1) . Since Q lies on the circle x^2 + y^2 = 1 , we substitute its coordinates into the circle's equation: (2t)^2 + (2t+2c-1)^2 = 1 4t^2 + 4t^2 + 4t(2c-1) + (2c-1)^2 - 1 = 0 8t^2 + 4(2c-1)t + 4c^2 - 4c = 0 Dividing by 4 , we get a quadratic in t : 2t^2 + (2c-1)t + c^2 - c = 0 For two distinct chords to exist, t