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Question

Tangent is drawn to ellipse x227+y2=1 at
(33cosθ,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


A

π3

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B

π6

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C

π8

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D

π4

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Solution

The correct option is B

π6


Given, tangent is drawn at (33cosθ,sinθ) to x227+y21=1.
Equation of tangent is xcosθ33+ysinθ1=1.
Thus, sum of intercepts =(33cosθ+1sinθ)=f(θ) [say]
f(θ)=33sin3θcos3θsin2θcos2θ, put f(θ)=0
sin3θ=133/2cos3θ
tanθ=13,i.e.θ=π6 and at θ=π6,f′′(0)>0
Hence, tangent is minimum at θ=π6.


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