Let a tangent be drawn to the ellipse x 2 27 + y 2 = 1 at ( 3 3 cos θ , sin θ ) where θ ∈ 0 , π 2 . Then the value of θ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Step-by-step solution

Equation of tangent be x cos θ 3 3 + y · sin θ 1 = 1 ,    θ ∈ 0 , π 2 intercept on x -axis O A = 3 3 sec θ intercept on y -axis O B = cosec θ Now, sum of intercept = 3 3 sec θ + cosec θ = f ( θ ) let f ' ( θ ) = 3 3 sec θ tan θ - cosec θ cot θ = 3 3 sin θ cos 2 θ - cos θ sin 2 θ = cos θ sin 2 θ · 3 3 ⏟ ⊕ tan 3 θ - 1 3 3 = 0 ⇒ θ = π 6 ⇒ at θ = &