Ellipse 2 Question 14

14. Tangent is drawn to ellipse x227+y2=1 at (3 3cosθ,sinθ) (where, θ(0,π/2).

Then, the value of θ such that the sum of intercepts on axes made by this tangent is minimum, is

(2003,1M)

(a) π3

(b) π6

(c) π8

(d) π4

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Solution:

  1. 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)=0sin3θ=133/2cos3θtanθ=13, i.e. θ=π6 and at θ=π6,f(0)>0

Hence, tangent is minimum at θ=π6.



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