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Question

The equation of the pair of straight lines through origin, each of which makes as angle α with the line y = x, is


A

x2+2xysec2α+y2=0

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B

x2+2xycosec2α+y2=0

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C

x22xycosec2α+y2=0

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D

x22xysec2α+y2=0

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Solution

The correct option is D

x22xysec2α+y2=0


Any line through the origin is y = mx.If it makes an angle α with the line y = x, then we should have

tanα=±{m1m21+m1m2}=±(m1)1+m

or(1+m)2tan2α=(m1)2

m22m{1+tan2α1tan2α}+1=0

m22msec2α+1=0, {1+tan2α1tan2α=sec2α}

But m=yx, hence on eliminating m, we get the required equation y22xysec2α+x2=0


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