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If n lines are drawn in a plane such tha...

If n lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent, such that these lines divide the plane in 67 parts, then find number of different points at which these lines will cut.

Text Solution

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Given number of straight lines=n, then
`1+underset(k=1)overset(n)(sum)k=67implies(n^(2)+n+2)/(2)=67`
`impliesn^(2)+n-132=0 implies(n+12)(n-11)=0`
`thereforen=11,n ne-12`
Hence, required number of points`=.^(n)C_(2)=.^(11)C_(2)=(11*10)/(2)`
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