Let P a sec θ , b tan θ and Q a sec ϕ , b tan ϕ where θ + ϕ = π 2 , be two points on the hyperbola x 2 a 2 - y 2 b 2 = 1 . If the ordinate of the point of intersection of normals at P and Q is - k a 2 + b 2 2 b , then k is equal to

Step-by-step solution

We have, x 2 a 2 - y 2 b 2 = 1       . . . i On differentiating w.r.t., x , we get 2 x a 2 - 2 y b 2 × d y d x = 0 ⇒ d y d x = b 2 a 2 x y ⇒ - d x d y = - a 2 b 2 y x Slope for normal at the point P a sec θ , b tan θ is - d x d y P = - a 2 b tan θ b 2 a sec θ = - a b sin θ ∴ Equation of normal at a sec θ , b tan θ is y - b tan θ = - a b sin θ x - a sec θ ⇒ a sin θ x + b y = a 2 + b 2 tan θ ⇒ a x + b cosec &