Let the tangent to the circle C 1 : x 2 + y 2 = 2 at the point M - 1 , 1 intersect the circle C 2 : x - 3 2 + y - 2 2 = 5 , at two distinct points A and B . If the tangents to C 2 at the points A and B intersect at N , then the area of the triangle A N B is equal to

Step-by-step solution

Let O 3 , 2 be the centre of the circle C 2 . The tangent to the circle C 1 :   x 2 + y 2 - 2 = 0 at M - 1 , 1 is x - y + 2 = 0 Now the perpendicular distance from O to the tangent is O P = 3 - 2 + 2 2 = 3 2 Also A P = O A 2 - O P 2 = 5 - 9 2 = 1 2 Now tan θ = O P A P = 3 ∴   sin θ = 3 10 = A P A N   &   cos θ = 1 10 ⇒ A N = 10 3 A P = 10 3 × 1 2 = 5 3 = B N Area of Δ A N B = 1 2 · A N 2 sin 2 θ = 1 2 × 10 9 × 3 5 = 1 6