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Question

If the equation of the lines passing through point (1,1), one making an angle θ with the positive direction of xaxis and the other making the same angle with the positive direction of yaxis, is x2(a+2)xy+y2+a(x+y1)=0,a2, then the value of sin 2θ is

A
a2
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B
a+2
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C
2(a+2)
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D
2a
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Solution

The correct option is C 2(a+2)
Let tanθ=m
Equations of the given lines are y1=m(x1) and
y1=1m(x1)
So their combined equation is
(y1)2+(x1)2[(x1)(y1)](m+1m)=0
x2+y22(x+y1)(m+1m)xy+(m+1m)(x+y1)=0x2+y2(m+1m)xy+(m+1m2)(x+y1)=0
Comparing with the given equation, we get
a+2=m+1m=tanθ+cotθ1sinθcosθ=a+2sin2θ=2a+2

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