If the straight line x cos α + y sin α = p touches the curve x a n + y b n = 2 at the point a , b on it and 1 a 2 + 1 b 2 = k p 2 then k =

Step-by-step solution

The line x cos α + y sin α = p is a tangent to x a n + y b n = 2 at a , b . Differentiating x a n + y b n = 2 , we get n a n x n - 1 + n b n y n - 1 d y d x = 0 i.e. d y d x a , b = - b a Now equation of tangent at a , b will be y - b = - b a x - a ⇒ a y - a b = - b x + a b ⇒ b x + a y = 2 a b Comparing with x cos α + y sin α = p , we get cos α = b ,   sin α = a ,   p = 2 a b i.e. 1 a 2 + 1 b 2 = 1 cos 2 α + 1 sin 2 α = 1 sin 2 α cos 2 α Also,