Let S be the set of all the natural numbers, for which the line x a + y b = 2 is a tangent to the curve x a n + y b n = 2 at the point a , b , a b ≠ 0 . Then

Step-by-step solution

x a n + y b n = 2 Differentiating, we get, n x a n − 1 ⋅ 1 a + n ⋅ y b n − 1 ⋅ 1 b ⋅ d y d x = 0 d y d x = - 1 a x a n − 1 1 b y b n − 1 = − b a x b y a n − 1 ⇒ d y d x a , b = − b a a b b a n − 1 = − b a So, equation of the tangent to the curve at a , b will be y − b = − b a x − a ⇒ y b − 1 = − x a + 1 ⇒ x a + y b = 2 i.e. S ∈ N