The area (in sq. units) of the part of circle x 2 + y 2 = 169 which is below the line 5 x - y = 13 is π α 2 β - 65 2 + α β sin - 1 12 13 where α , β are coprime numbers. Then α + β is equal to

Step-by-step solution

Given: x 2 + y 2 = 169 , 5 x - y = 13 ⇒ x 2 + 5 x - 13 2 = 169 ⇒ x 2 + 25 x 2 + 169 - 130 x - 169 = 0 ⇒ 26 x 2 - 130 x = 0 ⇒ x x - 5 = 0 ⇒ x = 0 , 5 ⇒ y = - 13 , 12 So, the points of intersection are 0 , - 13 and 5 , 12 . So, the required area is given by, A = ∫ - 13 12 169 - y 2 dy - 1 2 × 25 × 5 ⇒ A = x 2 169 - x 2 + 169 2 sin - 1 x 13 - 13 12 - 1 2 × 25 × 5 ⇒ A = π 2 × 169 2 - 65 2 + 169 2 sin - 1 12 13 Hence, on comparing with given value we get, ⇒ α = 169 , β = 2 ⇒ α + β = 171