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Question

If θ1 and θ2 be the angles which the lines
(x2+y2)(cos2θsin2α+sin2θ)=(xtanαysinθ)2 make with the axis of X, θ=π6, then tanθ1+tanθ2=ABcsc2α. Find the value of AB

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Solution

It is given that
(x2+y2)(cos2θsin2α+sin2θ)=(xtanαysinθ)2

(x2+y2)(cos2θsin2α+sin2θ)=x2tan2α+y2sin2θ2xytanαsinθ

(cos2θsin2α)y2+2(tanαsinθ)xy+(cos2θsin2α+sin2θtan2α)x2=0

Now, sum of the slopes =coefficient of xycoefficient of y2

tanθ1+tanθ2=2tanαsinθcos2θsin2α

Putting θ=π/6
We get, tanθ1+tanθ2=2sinα×1234sin2αcosα

tanθ1+tanθ2=83×2sinαcosα

tanθ1+tanθ2=83csc2α

On comparing this equation with given equation, tanθ1+tanθ2=ABcsc2α

We get A=8 and B=3

AB=5

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