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If t(1),t(2) and t(3) are distinct, the ...

If `t_(1),t_(2) and t_(3)` are distinct, the points `(t_(1)2at_(1)+at_(1)^(3)), (t_(2),2"at"_(2)+"at_(2)^(3)) and (t_(3) ,2at_(3)+at_(3)^(3))`

A

`t_(1)t_(2)t_(3)=1`

B

`t_(1)+t_(2)+t_(3)=t_(1)t_(2)t_(3)`

C

`t_(1)+t_(2)+t_(3)=0`

D

`t_(1)+t_(2)+t_(3)=-1`

Text Solution

Verified by Experts

The correct Answer is:
C
The given point are collinear if
`|{:(t_(1),2at_(1)+at_(1)^(3),1),(t_(2),2at_(2)+at_(2)^(3),1),(t_(3),2at_(3)+at_(3)^(3),1):}|=0`
`rArr|{:(t_(1),2t_(1)+t_(1)^(3),1),(t_(2),2t_(2)+t_(2)^(3),1),(t_(3),2t_(3)+t_(3)^(3),1):}|`
Applying `R_(2)rarrR_(2)-R_(1),R_(3)rarrR_(3)-R_(1)"we get"`
`rArr(t_(2)-t_(1))(t_(3)-t_(1))|{:(t_(1),2t_(1)+t_(1)^(3),1),(1,2+t_(2)^(2)+t_(1)^(2)+t_(2)t_(1),0),(1,2+t_(3)^(2)+t_(1)^(2)+t_(3)t_(1),0):}|=0`
`(t_(2)-t_(1))(t_(3)-t_(1))(t_(3)-t_(2))(t_(3)+t_(1))=0`
`rArr t_(1)+t_(2)+t_(3)=0`
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