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Question

If the line segment joining the points A(a,b) and B(c,d) subtends an angle θ at the origin, then cosθ is equal to

A
ab+cd(a2+b2)(c2+d2)
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B
ab+bd(a2+b2)(c2+d2)
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C
ac+bd(a2+b2)(c2+d2)
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D
None of these
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Solution

The correct option is B ac+bd(a2+b2)(c2+d2)
The equation of the line through the origin and (a,b) is y=bax......(1)
And the equation of the line through the origin and (c,d) is y=dcx.......(2).
Given θ be the angle between (1) and (2).
Then
tanθ=dcba1+bdac
or, tanθ=adbcac+bd
Then cosθ=ac+bd(adbc)2+(ac+bd)2
or, cosθ=ac+bda2d2+b2c2+a2c2+b2d2
or, cosθ=ac+bd(a2+b2)(c2+d2).

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