Tangent is drawn to ellipse   x227+y2  =  1  at  (33 cosθ,  sinθ) (where θ∈  (0,  π/2)) . Then the value of such that sum of intercepts on  axes made by this tangent is least is

Equation of tangent at (33cos⁡θ,sin⁡θ) is x cosθ33  + y   sinθ  =  1Its intercept on coordinates axes are  33/cosθ  and  cosec  θ If S denotes their sum, thenS=33sec⁡θ+cosec⁡θdSdθ=33sec⁡θtan⁡θ−cosec⁡θcot⁡θ For minimum value of S⋅dSdθ=0⇒ 33sin⁡θcos2⁡θ−cos⁡θsin2⁡θ=0⇒tan3⁡θ=33 Since θ∈(0,π/2), so tan⁡θ=3⇒θ=π/3 Also d2Sdθ2=33sec⁡θtan2⁡θ+33sec3⁡θ+cosec2⁡θcot⁡θ+cosec3⁡θ⇒d2Sdθ2θ−π/3>0.  Thus S_ is least when θ=π/3