Let A B C D be a square of side of unit length. Let a circle C 1 centered at A with unit radius is drawn. Another circle C 2 which touches C 1 and the lines A D and A B are tangent to it, is also drawn. Let a tangent line from the point C to the circle C 2 meet the side A B at E . If the length of E B is α + 3 β , where α , β are integers, then α + β is equal to ________.

Step-by-step solution

According to the question we get, Here, A O + O P = 1 or 2 + 1 r = 1 ⇒ r = 1 2 + 1 × 2 - 1 2 - 1 = 2 - 1 . The equation of circle is x - r 2 + y - r 2 = r 2 . The equation of tangent C E of slope m is y - 1 = m x - 1 ⇒ m x - y + 1 - m = 0 It is tangent to the circle, then the distance of center of the circle C 2 r , r from the tangent C E is equal to the radius of circle r . ∴   m r - r + 1 - m m 2 + 1 = r ⇒ m - 1 r + 1 - m m 2 + 1 = r ⇒ m - 1 2 r - 1 2 m 2 + 1 = r 2 Put r =