Tangent is drawn to ellipse x227+y2=1 at
(3√3cosθ,sinθ) (where,θ∈(0,π2)).
Then, the value of θ such that the sum of intercepts on axes made by this tangent is minimum, is
π6
Given, tangent is drawn at (3√3cosθ,sinθ) to x227+y21=1.
∴ Equation of tangent is xcosθ3√3+ysinθ1=1.
Thus, sum of intercepts =(3√3cosθ+1sinθ)=f(θ) [say]
⇒f′(θ)=3√3sin3θ−cos3θsin2θcos2θ, put f′(θ)=0
⇒sin3θ=133/2cos3θ
⇒tanθ=1√3,i.e.θ=π6 and at θ=π6,f′′(0)>0
Hence, tangent is minimum at θ=π6.