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Prove that the lines ax+by + c = 0, bx+ ...

Prove that the lines `ax+by + c = 0, bx+ cy + a = 0 and cx+ay+b=0 ` are concurrent if `a+b+c = 0 ` or `a+b omega + c omega^(2) + c omega = 0 ` where `omega ` is a complex cube root of unity .

A

`a+b+c=1`

B

`aomega+bomega^(2)+c=0`

C

`a+bomega^(2)+comega=0`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Since In `x AM` and `ND` are concurrent `|(a, b, c),(b, c, a),(c, a, b)|=0`
`a(bc-a^(2))-b(b^(2)-ac)+c(ab-c^(2))=0`
`abc-a^(3)-b^(3)+abc+abc-c^(3)`
`3abc-a^(3)-b^(3)-c^(3)=0`
`a^(3)+b^(3)+c^(3)=3abc=(a+b+c)(a+bomega+comega^(2))(a+bomega^(2)+comega)=0`
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