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Question

If θ1 and θ2 be the angles which the lines (x2+y2)(cos2 θ sin2 α+sin2θ)=(x tan αy sin θ)2 make with the axis of x, then if θ=π6, tan θ1+tan θ2 is equal to
  1. (83)cosec 2α
  2. (83)sin2 α
  3. 83 cosec 2α
  4. 4 cosec 2α


Solution

The correct option is A (83)cosec 2α
The given equation can be written as
(x2+y2)(cos2 θ sin2 α+sin2 θ)=x2 tan2 α2xy tan α sin θ+y2 sin2 θ or (cos2 θ sin2 α+sin2 θtan2 α)x2+2(tan α sin θ)xy+cos2 θ sin2 αy2=0
Since the slope of these lines are given as tanθ1 and tan θ2.
and sum of the slopes =2 tan α sin θcos2 θ sin2 α
   tan θ1+tan θ2=2 tan α2×(34)×sin2 α   (  θ=π6)=(83)cosec 2α

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