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Q.

A tangent is drawn at (33cosθ,sinθ)0<θπ2 to the ellipse x227+y21=1. The value of θ for which the sum of  the intercepts on the coordinate axes made by this tangent  attains the minimum is

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a

π6

b

π3

c

2π3

d

2π4

answer is A.

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Detailed Solution

Given ellipse x227+y21=1

Equation of tangent at (33 cosθ, sinθ) is

 x(33 cosθ)27+y(sin θ)1=1 x33 secθ+ycosecθ=1

sum of intercepts =33 secθ+cosecθ

lLet f(θ) =33 secθ+cosecθ Now f1(θ)=33 sin3θ-cos3θsin2θ cos2θ

for max or min f1(θ)=0

33 sin3θ=cos3θ tan3θ=133=133 tanθ=13=tanπ6 θ=π6

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A tangent is drawn at (33cosθ,sinθ)0<θπ2 to the ellipse x227+y21=1. The value of θ for which the sum of  the intercepts on the coordinate axes made by this tangent  attains the minimum is