+y2=1 Now, equation of tangent to the ellipse at P(3√3‌cos‌θ,sin‌θ) is given by ‌
3√3‌cos‌θ⋅x
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+sin‌θ⋅y=1 ⇒‌‌‌
x‌cos‌θ
3√3
+y‌sin‌θ=1 ...(i) X-intercept of Eq. (i) is x=3√3‌sec‌θ=OA (say) Y-intercept of Eq. (i) is y=cosecθ=OB (say) ∴ Sum of intercepts =3√3‌sec‌θ+cosecθ=f(θ) (say) ⇒f′(θ)=3√3‌sec‌θ‌tan‌θ−cosecθ⋅cot‌θ Put f′(θ)=0
⇒‌‌‌
3√3‌sin‌θ
cos2θ
=‌
cos‌θ
sin2θ
⇒‌
sin‌θ
cos2θ
⋅‌
sin2θ
cos‌θ
=‌
1
3√3
⇒‌‌tan3θ=‌
1
3√3
⇒θ=
Ï€
6
∵f′(θ) changes sign from negative to positive when moving from left to right. ∴f(θ) will attain minima at θ=‌